The tricky part is that human beings are very complex, and very finicky about their environment. As noted in answers to your previous question, there's no simple way to go from a bit of DNA to a full-grown human without some serious future-tech machinery. And since you noted that your humans are barely capable of starflight, they probably can't create living humans just from their DNA, much less having it happen by accident.
So, if we can't rely on something that exists on the human ship, let's assume our solution comes from the planet itself. I'll go with the weaker part of the question (habitable planet with simple life), since it makes everything else a bit more plausible :)
The human ship was an Ark. It contained samples of DNA of many of Earth's life forms, selected specifically in an attempt to recreate an Earth-like ecosystem (of course, a much simpler version of it). Plants to capture carbon dioxide, water and sunlight in Earthly sugars. Bacteria and algae to produce the main source of oxygen, and seed the oceans with nutrients. Another bacteria and funghi to create a soil cycle. The bare essentials to produce a world that's habitable for humans. As a part of this, a bunch of seed environments were ready on the Ark, to kickstart the whole thing, and cryo chambers to let the supervisors manage the project over the scope of many human lifetimes. Sadly, the ship crashed, and everyone on board has been killed. Well, everyone, except for the seed environments - and the primitive Earth organisms get sprayed all around the Planet. Most of them survive, and replicate - it so happens that they are much better at what they do than all of their native competitors. The environment starts filling with Earthly sugars, amino acids, fats, proteins... easily accessible raw materials, which happen to not be compatible with the existing Planet life.
The Planet life is rather primitive in Earthly terms. They have many similiarities, but aren't able to process Earthly proteins and fats at all, and even the sugars give them trouble. DNA is too stable to be broken apart by the locals. And as the ecosystem quickly (over thousands of years) becomes dominated by Earthly "raw" materials (remember, the Earthly life is much better at processing material and sunlight, and easily outcompetes the locals), Planet's indigenous life is in jeopardy, as its entire food supply starts to quickly disappear. Many of the locals tried to digest the Earthlings, but with no success - leaving the invaders inside the cells and bodies of the locals, neither being able to consume the other. Ultimately, it only served as a way to spread the Earthlings further. Until...
Finally, under just the right environmental conditions, a small portion of the indigenous population managed to break through the cells of the invaders. The two organisms effectively merged together - their cell boundaries broken, the locals enslaving the Earthlings to produce energy for them (it so happens that both the Planetians and the Earthlings use ATP for cell energy transfer), to hijack on their ability to process Earthly materials, and to more effectively capture sunlight. A tiny portion of the local portion manages to unwrap the invader DNA in just the right places, to use the invader ability to break down and synthesize proteins, and a tiny portion of those manage to incorporate the cell organelles of the invaders in their own cells. The tiny population quickly spreads all over the planet - while it's not much better than the invaders, it can at least fully exploit food sources of the invaders; keeping close check on their own genetic information, while preserving the useful bits from the Earthling's. Since they only use rather small portions of the Earthling genetic code, there's significant pressure to avoid replicating the "unused" bits, but it is more than counter-acted by every tiny trick that is found in the previously unused bits. Most of the time, the random unwrapping results in waste, or even in the death of the organism - but as the new tricks improve the adaptability of the locals, they quickly grow in number. The result is a somewhat pruned genetic information of the Earthlings, taking over more and more functions in their "host" organism. And then...
The DNA vials from the Ark start leaking. The locals, ravenous for any new Earthling genetic information, start absorbing the information, replicating it with them as they always do, trying many different ways of unwrapping it, and unlocking the processes described in the information. Predictably, this is met with little success - it's very unlikely that you'd get a viable organism just by randomly trying to execute various parts of their genetic code, even when most of yourself has already transformed to something very similar to Earth life. But while the chance is tiny, it isn't zero. And finally, it happens - the right kind of mixed local-Earthling happens to unlock the right part of the "stolen" DNA... and result in horrible outgrowths, nothing like the locals have ever seen - the Earthly processes using all the infrastructure of the surrounding local cells to spark the beginning of Earthly plant life, sprouting as grasses and funghi and bushes and trees...
The local meta-cells are quite adaptable, working with many different kinds of Earthly DNA. Most of the time, the synthesis fails, but the invader's DNA usually stays the same throughout, giving many chances at repeated attempts to recreate Earthly life. More and more complex life starts getting hold, relying on their host Planetians to spread their kind before their own replication have a chance of working. The relationship is quite symbiotic - even as the different kinds of symbiotes start devouring each other, funghi reclaiming the dead (and sometimes not so dead) bodies of the plants, Earthly soil bacteria finally getting all the nutrients they need, fixing more and more nitrogen and other nutrients in the soil, quickly spreading more and more advanced life forms all over the Planet.
Even plants are a huge stretch of imagination, the conditions for a creation of a viable seed-organism being extremely unlikely. But there's trillions of the native meta-organisms, each trying with their own copies of Earthly DNA, merging together in huge colonies in an effort to produce a viable Earth-organism, and they do have the plans - they just have to find the right conditions, the right sets of instructions to execute in just the right way, and over time, it happens, and enriches the environment. Eventually, there's enough of Earthly species to start reproducing on their own, spreading in a race for diversity to cover all the ecosystems of Planet, creating thousands of new species. And then...
Come the insects. The native meta-organisms don't stop trying, keep reiterating over the absorbed DNA, mix together with other organisms, Earthly and local, and eventually, manage to build a functioning reproductive system in a meta-plant, capable of producing insect younglings; not quite like they would be in their natural environment - with no adults to take care of them, they rely on their home-plant for shelter and food, until they are fully grown and capable of raising the next generations, and producing viable reproductive individuals, which quickly spread all over the Planet - some still clinging to their home-plants, while others are barely different from their Earthly brethren.
But while much more complicated than anything that came before, the conditions to grow a mammal or a reptile from "scratch" are even more so. While everything necessary to create the organs is in the genetic information, it's mind bogglingly unlikely they would be executed in combined in just the right way to produce the necessary environment. It's quite unlikely that there would be enough of the original DNA left, given how much of a waste it is to replicate large swathes of information that is never used - but then again, the same is true for our own junk in our own DNA; full information survives, just because the random changes to make the information shorter are very likely to break the parts that are necessary. It finally happens - home-plants with full blown wombs and egg-production-facilities, relying on their newly formed animals to spread their own seed.
The first humans are rather troublesome. They are huge, requiring lots of nutrients to grow. When they leave their reproduction chambers, they are all but hopeless - and the meta-organisms have no way to help them; there's no way to use the information they have to help with the maturation of a human. Still, many human-plants survive, by using the babies as carriers of their seed, the "fruits" becoming popular enough with the "predators". But by then, simpler mammals have already proliferated enough. And mammals have this weird connection, the "cuteness" factor, which works quite well accross species... simpler species nurturing the more complex ones, which in turn become populous and ready to nurture even more complex ones... all the way to the wolf that takes care of the helpless human twins, nurturing them and caring for them until they are themselves capable of taking care of their own (from other human-plants), forming a symbiotic relationship with the wolves that would last for ages - becoming surprisingly strong as they mature, something that Planet has never seen before...
... and thus Rome is born.
Note
I wouldn't call this the hardest sci-fi around. There's nothing here that would be outright impossible, but it's incredibly unlikely that something like this would happen. There's a lot of points where you deal with fragile things that survive against all odds - the DNA not deteriorating before it gets used, not being pruned by the natives, the fact that none of the local populations started becoming dominant as something else than the meta-organisms that recreate Earthly life...
But it should be quite enough to support an entertaining story without too much suspension of disbelief. It gives interesting opportunities to show clashes of the local and the Earthly. There's plenty of space for interesting organisms (and especially complex symbiotical relationships, given how central symbiosis is to the whole scenario) that quickly diversified in the rather empty ecosystem.
