So I guess my question is how do you deal with depressed individuals?
and how do you stop their mood from getting to you?
Now, for a less facetious answer: you should only do something if a colleague's behaviour becomes a distraction to the point where it is interfering with your work. In your situation, those sighs and complaints might rise to that level, in which case your first action should be to ask the person directly if he can tone it down. Your tone should be respectful but how you word that depends on your culture, relative position in the hierarchy and a lot of other factors.
If you've tried that many times without effect, you might have to learn to live with the behaviour or. If it's truly a disruption, you can escalate this to your manager who may or may not do something about it, or HR where the same caveat applies. Only go to HR if you know they're receptive to this kind of thing and have shown competence in the past. When you're raising this, focus on the behaviour, how it impacts you and remain matter-of-fact and non-judgemental. Explain that you've tried to work it out with the individual directly but that it's had no effect.
In fact, you should treat this as general "my coworker's behaviour is annoying me" situation. That question has been covered here before: What can I do about a very loud coworker?
As for his dislike of you, your coworkers don't have to like you to maintain a professional relationship. Jealousy and bigotry are personal defects that say much more about this colleague than about you. Write him off as a boor and remain professional but distant. If his behaviour is more severe than you describe that is a different problem and there are a fair number of question on the site about dealing with such a situation.
As for the broader question of depression: a person's mood, medical history or psychological condition is really no concern of yours. Guessing at a diagnosis is pointless: you should just focus on a person's behaviour and aim to preserve a good working relationship. About the only exception to this is when a person is exhibiting signs of compromised functioning or severe medical symptoms: expressing suicidal thoughts, delusions, suddenly slurred speech and so on. But such signs are typically very recognisable and they're a far cry from someone feeling frustrated or a bit melancholic at work.