I think the title speaks for itself. The species I'm designing doesn't have a sense of greed like us (greed isn't part of their nature). In fact, they deny the existence of something like greed, because it's simply not a part of their world. But back to the quiestion, can a species continue to develop as a civilization if it didn't have the sense of greed?
Necessity is the mother of Invention, but curiosity drives exploration.
In your setting, you've stipulated that the NEED for space travel is not presented, for instance by thinning resources on the planet, but with intelligence comes curiosity, so at some point, an inevitable urge to examine the space above their heads.
To me, this is enough to suggest that even without a competitive drive, there would still be sufficient motive to push a civilisation to reach for the stars.
As for the second aspect of your question, could an ecosystem exist around a purely co-operative model with no trace of competition, well, I find this unlikely. You see, even in our nature there are many examples of co-operative evolution. Plants developed flowers of varying colours to attract creatures to them which in turn prompted the evolution of insects and birds that specialised in pollenating these flowers, however there was still competition among the species to find the best, most attractive flowers, to have the optimal strategy.
At some point it's not good enough to just be able to co-operate or face extinction, you still have to co-operate better than the other species in your class, or get left behind, and die out.
To remove the natural consequences that the strongest and best adapted species survive means that you are "carrying" every species that exists.
Looking at predator and prey balance, the kind of co-opeartaion you're suggesting would almost require a sort of ecological social welfare system. A species with no natural defences, low speed and reflexes and a very low rate of reproduction is meant to be ignored for the easy foodstuff it makes for the predators in the area? It would likely get hunted to extinction, in absence of any motive for it's predators to eat anything else until it is gone, unless the predators all agreed to conserve the last remaining populations of that species purely for the purpose of saving it from extinction. This does not serve any evolutionary purpose.
I think the closest reality to the one you are suggesting would be one that focusses survival on the efficiency of co-operation, that all things work in a symbiosis, but evolution is still driven by competition to be the best co-operator, that guaranteeing your usefulness keeps your species alive, until another species proves to be more useful, then it's back to adapt or die.
Yes, this already happened
I think your notion that humans are inherently competitive is inaccurate, at best. 'Competitive' and 'cooperating' aren't two binary options, it's a false dilemma.
No species will ever be 100% competitive nor 100% cooperative, but somewhere between those extremes. Humans certainly aren't 100% competitive. If anything, we are more cooperating than many (most?) other species. You don't go around killing people for food, right? In fact, chances are that you have a job and work with other people, producing something of value, while someone else produces the food you eat, the electricity you consume and so on. Look around you, think about how many people, all over the world, engaged in cooperation to make the stuff you see. And this is natural to humans.
Back to your question, I find it hard to believe that a theoretical 100% competitive species will establish a civilization. Likewise the 100% cooperative one. Extremes will never get you anywhere.
But, excluding purely theoretical extremes - everything is possible. Fact is, we do not have the slightest idea which qualities are required to establish a space-faring civilization - as we only ever saw one successful example (from one single planet).
Aside from some far-fetched speculation, I cannot see a reason why would a somewhat more cooperative civilization than humans fail to reach space.
Worth noting that there is a lot of information lacking in the question. from How big is the planet to what sort of landmasses and how the entire ecosystem co-operates. and how you define competition. and i'll ignore the predators and prey not competing as it makes my head hurt
But if... the planet was Earth. and the aliens were basically humans that were a lot nicer to each other and wildlife in general. then unfortunately no, not a chance.
For a minute lets ignore everything that comes before space travel, all the research, all the technological developments that would be required to reach the point that Wernher Von Alien says "hey, i reckon i can put us in space." Lets also ignore the origins of space travel for humans, as it was the biggest game of oneupmanship ever, between Wernher Von Alien and Sergei Koralien
What is required to reach space: a massive rocket.
When you consider that rockets require computer guidance systems, rocket fuel, a big tube to put the fuel in, and some rocket engines. Yes this i very much an oversimplification!
Now what does all that that require. a lot of very refined material. gold, tungsten, Steel, Copper... so so much copper, a absolutely huge amount of steel, and literally tonnes of aluminium, and a lot of other materials in varying amounts.
So if the entire planet lived in harmony. where does this material come from, at some point something will have to make way as the United States of Aliens compete with wildlife for the regions rich in these materials in order to build the rocket, some sort of wildlife will have to move out of home. this is competing for land against wildlife. or maybe even competing for land against others of the same intelligent species.
Then of course, when the rockets fire, what happens to the debris in the event of an accident, it WILL damaged someones back garden, whether that be actual people, or land animals or fish, someone will have to deal with the damage or debris, so how does that work. sorry guys we've decided to go to the moon, and that chunk of metal that just landed on and crushed those crabs... it was for the betterment of the entire planet.
The concept is great for primitive species, Movies like Avatar play this theme quite well. but advanced civilizations do require for them to compete against nature. it would definitely be possible for a peaceful species to acheive spacetrave while taking care not to damage too much of the planet, but one where everything is harmonic. i just don't think its possible
I don't believe we have enough information to answer the 'will they be able to achieve spaceflight' question beyond a simple 'not as you describe it'. We need a lot more information about how your world actually functions before answering such a detailed question.
So, let's answer your 'can this work' question (which I think is a better starting point). Bear with me, this might be a long one, but hopefully it's worth it!
Can a world based on co-operation not competition exist?
Unfortunately, the answer is still 'not as you describe it'.
The issue is one of simple efficiency. In the vast majority of cases, it is more efficient to compete with other species for limited resources. From an evolutionary perspective, every resource is limited. If there is a surplus of resources, species will increase in numbers until a critical resource is no longer in surplus. This is most likely the reason competition is almost ubiquitous in our ecosystem. However, I do think that we might be able to engineer a set of circumstances in which an ecosystem of co-operators might exist.
In order to start, we need to define the main types of competition from the perspective of a species. Firstly, we have interspecies competition. Two species both need a resource, so compete with each other to access it. Secondly, we have intraspecies competition. Two members of the same species both need a resource, so compete with each other to access it.
Let's look at interspecies competition first. In order to reduce this to negligible levels, we need to engineer a situation in which two species are able to support larger numbers by co-operating rather than competing. This is common in our world (extreme forms being symbiosis, but plenty of lower-level co-operation. If necessary I'll reel off a list, but only if necessary).
What's rarer, and more difficult to justify is predator-prey symbiosis. Surely if something wants to eat you, you're going to compete with it to not be eaten. The question becomes 'how can we make it more evolutionarily beneficial to allow yourself to be eaten by a predator?
Luckily for us, we've already got a few examples of animals allowing themselves to be eaten. In some praying mantises and some species of spider, the males allow themselves to be eaten by the females after mating. This gives the females a much-needed food source immediately prior to producing their mutual offspring. So, what we need is a way for a prey-animal to have a vested interest in the survival of their predator.
How about larval tapeworms? Lets say herbivore A reproduces by infecting predator B with tape-worm young (evolutionary path in the post-script). Now, hebivore A has a vested interest in the survival of its predator as it's now carrying its young. We have a lifecycle that involves a prey animal specifically seeking out another animal to eat it.
Energetically speaking, this is less efficient than just letting your larval young eat you. So we need another example to help guide us.
What about fruit? Ignoring the whole animal/plant divide (it's very much an earthling thing), we can define 'predation' as 'one organism subsisting off another organism'. If an animal eats fruit, it's definitely eating another organism. And yet trees go out of their way to make fruit really edible.
The reason they do this is because it's more efficient at propagating their seeds, spreading them far and wide, avoiding competition with the parent tree. As we'll see in a bit, this isn't a great example but what it does do is establish that there are other methods of generating efficiency at propagation than 'survive to create more babies'. Either that, or we could modify herbivore A so that they drop a highly-nutritious larval sac when they're scared, allowing them to survive to breed again.
It's a bit of a stretch to expand that method of reproduction to the majority of species on a planet, but evolution is strange with things like that. If one lucky larval-procreator survived some ancient extinction event and then speciated to become the dominant class of animals on a planet (somewhat of like mammals), then we might end up with a Planet of Hats where the 'hat' is interspecies co-operation and symbiotic tapeworm-larvae reproduction.
This is sounding like less and less like an ideal holiday destination...
I've got a blurb to go with this as well, but I've already wasted too much company time writing this rambling diatribe. My idea, once I've edited it, hinges on high-eusociality and low genetic difference meaning it's more efficient to allow others to procreate when you could otherwise have done.
- Initially, I'd imagine its larvae would just strike it out in the world like everyone else's larval young, hoping to infect another animal. But why waste energy looking for animals to infect when the animals can come to you? Heading through a stage where scavengers get infected after feasting on herbivore-A carrion, we get to a point where herbivore A can reproduce faster if it has a high turnover of generations feeding a predator that constantly maintains a colony of fast-maturing larvae of another species. Win-win. Herbivore A gets to offset the costs of pregnancy a little, and predator B gets a steady diet of tasty larvae (bonus points if they time their suicide to a point in time when there's few other prey-animals available so predator B becomes dependent on our little lemmings for survival).
tl;dr Turns out co-operation-world Utopia 1 is dominated by eusocial horrors that reproduce through infecting their predators with semi-symbiotic tapeworms. We don't recommend sampling the local cuisine, no matter how keen they are for you to try it.
It's a very interesting question. If I may, I'll start by speculating about where you're coming from on this -- or at least, what it makes me think of!
1. Some motives for the problem?
For a long time, the standard trope in sci fi was aggressive, militaristic aliens invading earth. Later, it was asked "why are these supposedly advanced civilizations still showing primitive, aggressive impulses like us humans?" And the aliens instead became benign beings, possibly trying to guide us on our own path of higher development.
More recently, people have started to suggest that the earlier model was in fact more realistic. The driving impulses in our species that gave us advanced civilisations and technology, and the possibility of becoming a space-faring species, all arise out of our aggressive past:
- Predators tend to be much more intelligent than herbivores, filter-feeders, and so on. (You don't need brains to stalk grass.) 
- Pack predators develop the degree of co-operation that enables the group to achieve much more than the sum of its parts, whilst maintaining an intense competition for resources which drives both biological and technological evolution
So in fact it seems pretty likely that the first alien civ we meet will be more like us: a thin veneer of civilization pasted on top of a vicious pack predator.
2. But why?
The fundamental drivers that create this are rules of evolution and ecology that are not even specific to, say, DNA based life; they arise from the very pattern of which life is created: individuals that each contain a genome, which survives through time by replication and consumption of resources. Anything that follows the same basics will develop similar rules. For example, the correct balance betwen co-operation and competition is found even in very simple, artifical games like Prisoner's Dilemma; and it is seen to evolve in highly artificial systems like Game of Life.
3. So are there any other options?
This is where it gets tricky. You are asking us to imagine something that is not only unlike any other life we know, but anything that follows the same general pattern of life -- and yet is still, in some sense, alive. I have two very incomplete offerings:
- Life has evolved under very different rules because it is being monitored and "gardened" by some external force which imposes those rules; or
- An ecosystem which in fact consists of a single mega-organism, where the competing organelles are regulated (similarly to option 1) by a central control mechanism.
How could either of these occur? I have no idea, sorry!
- It is exceedingly difficult to compare intelligence between widely different species. They have different sensory capabilities, different modes of movement, different motivations. So there is no such thing as an official zoologists' list of top N most intelligent species -- and many such lists that are published on the 'net are more to shock and surprise, than to inform. (No; sheep are not particularly intelligent. For a large mammal, they are pretty stupid, actually.) But the only herbivore that regularly figures highly is the elephant. Some parrots also figure pretty highly, notably the African Grey -- but the most intelligent parrots are not strict herbivores. The African Grey eats fruit, but also hunts insects, and probably relies on them during dry seasons.
Let's assume that a truly cooperative, all-life-loving species ever made the step to become comparable to humans in intelligence and technology. I think there are some vital ingedients to space exploration missing.
1. The incentive
Why would such a species want to reach space? You could argue with the general spirit of exploration or maybe some religious wanting to be near their God.
But realisticly, the spirit of exploration was fueled by competition. Explorers wanted be the first to see some foreign culture or to reach the top of a mountain. As soon as someone claimed the title of "First person to do X", the general desire to repeat the feat was low.
For religious enthusiasm the exact opposite often applies. Holy mountains are often so sacret that people are not allowed to climb them or ferain from doing so out of respect for the gods.
2. The idea
The first things that ever flew were insects, saurians and birds. They need an atmosphere to fly in and will never reach space.
The first things that could technically reach space were gunpowder propelled rockets build by the Chinese in the 13th century for military purposes. These simple, unguided contraptions were further developed in Asia and Europe and later by every military nation in the world, always for military purposes.
Konstantin Tsiolkovsky (1903) first speculated on the possibility of using rocket technology for manned spaceflight (Wikipedia)
That's a damn long time of military development until someone publicly voiced the idea of using a rocket for something different than killing. How and why would your cooperative aliens develop any idea about how they could transport someone into space if they never saw a rocket fly miles into enemy territory?
3. The development
Even if your aliens developed gunpowder and rockets to celebrate new years eve with bright sparks in the sky, how long would it take them to go from compackted gunpowder glued to a stick that sends sparks into the sky to a giant metal tube packed full of technology and volatile chemicals that sends one of them into the sky?
Many great inventions failed because they couldn't be implemented in a reasonable amount of time and ran out of interest and funding. The gap in technologies is just too big to bridge without actually having a use of the technology. We humans were only able to send a man into space because we sent many men into many wars.