They mean the same thing. I think the first on sounds more polite.

I hope we don’t lose any animals. It’s not wrong to be redundant, as you are in your sentences, but in general there should be a reason for using words in English. It’s always a good idea to think about the simplest and clearest way to say things and then ask yourself what’s gained by adding complexity and redundancy.

Well, they pretty much mean the same thing. "...are not going to lose..." is a future progressive verb phrase hinting at a process of losing animals, rather than flat negation of "lose" in the simple future case of "won't". However, "going to" or "gunna" has gradually taken over simple future tense, outside cases of emphasis: "We WON"T lose any animals in the future!" An English speaker is going to understand these sentences as pretty much being the same.

I agree with Glenn that "will not" is simpler and sounds better. I would also suggest ditching "hopefully". It is more straightforward to say: "I hope we don't lose any animals in the future." You can say "won't" instead of "don't" but "don't" sounds better. There really is no reason to use "will". "Will" is used to express certainty but there is no certainty here so "don't" works better.