Can a species devoted to destroying other species survive and prosper?
Yes. We are already have something similar.
Why does not a nuclear state simply nuke it's enemies?
Hmmm. Because of self survival and a complicated web of politics that leads to the fact that on the global level you can be destructive but not self destructive.
That web serves most countries involved even if you are getting the short hand.
Why do we care for nature to begin with?
Because without a decent level of nature we will start having trouble to the point of endangering us.
So most of humanity hold off the destruction aspect because we prioritize our own existence.
Yes others like to protect things and preserve them but does not matter here.
So basically we already exist in a state of do whatever you want as long as it continues your own existence, or the advancement of your ideas or offspring.
I know it is almost insane to compare or prove.
Our motives here could be to drink pineapple juice or rule the stars. The human ideologies and practices and history is a continuation of the same basic principle of self preservation.
Only we are smarter and so we extend that to other things such as abstract concepts and helping others...etc.
So again, we don't destroy things because we are good or bad or have good intentions or bad. We don't destroy things because we are smart enough not to do so.
Think of how we experiment on animals. The value of doing so is much higher than that pesky little morality of some people so hey valuable data.
Ask any general if they had the ability to test weapons on an alien planet with alien species without it being a war crime and tell his that the date will be 100% simulating the human battlefield what will they do?.
His disappointment of knowing that he can't will be immense.
Thus if you want to include whatever reason for "reckless hate" against everything else you absolutely can.
But intelligence will tune down the destruction based on consequences and so you will have more destruction but intelligence will insure the ultimate survival and thriving of the species.
This assumes that intelligence is hardcoded and their thinking hierarchy is basically like humans only they are more destructive but not insanely so.
Basically if you ask a random member of them why do you guys have laws regarding protection of nature he will tell: "Sure I'd like to burn the whole planet. But we know that without a certain level of protection then we will eventually die.
So once we can exist without need for nature I'll be the first to burn this mother"
Can they keep dogs? Maybe. They are useful and they can be, the species, crueler masters. But even dogs will only tolerate cruelty to an extent so they will push it further but only to a certain point.
Wait. Does that mean that they will learn the lesson that they can't do whatever they want?
Que Doctor Manhattan saying "I can change almost anything...but I can't change human nature." With the name of the species instead of human
Will they end up changing because of selective malice?
Again this is difficult as this little species of homo sapiens is like a quarter of a million years old. We don't know how we will evolve or not so that's a tough one.
So to sum it up: intelligence is the biggest factor here and as long as they are smart they will insure their survival while perusing secondary goals.
Ours is playing Skyrim and eating hamburger while theirs is torturing little kittens and burning trees. But neither changes the grander scheme of things.
To how far can you push it before your point of malice changes things?
I'd argue if they follow our evolutionary tree not much.
We breed for intelligence and it is the factor that made us what we are.
I'd even argue that pure and destructive nihilism is a sort of intelligent thought and thus you can't have it before having the basic intelligence level of self preservation on the larger scheme like humans.
I hope the bolding of passages is not random and that I answered the question.
I think our artistic examples only show this.
The Joker was smart enough to oversee his own plans and demonstrate it.
Khorne from warhammer 40K only cares for the blood flowing but insures that his subjects are given powers to burn the galaxy.
The Necrons, same universe, also corporate and plan and execute maneuvers and do all sorts of technological things and repair to insure their dogma of destroying all organic material is fulfilled.
Even the Reapers from Mass Effect do something similar and don't even target lesser intelligence species.
Those examples are the best I could come up with now. So if they don't fit then I just wanted to show my point.
Of course context rules here so if you have a species with specific properties then things might be different.