I'll start with the last two questions because they're less complicated:
A) what does it feel like to stay abroad?
B) what would it feel like to stay abroad?
"What does it feel like?" is something you'd ask somebody who has done the thing in question: in this case they've been abroad, so they know first hand what it does feels like. Whereas "What would it feel like?" is more hypothetical. It's asking you to imagine what it is like to stay abroad when you haven't done so, or to say how you think the person asking might feel if they stayed abroad; you don't know, because you haven't done it, but you can guess based on other experiences that you've had, and from what other people have told you.
Having said that, in many situations (especially informal ones), most people aren't super careful about their choice of words, so both questions could be interpreted the same way if they're asked of somebody who has not been abroad.
C) What does depression feel like?
D) What does it feel like to have depression?
E) What would it feel like to have depression?
All three of these could be interpreted as meaning the same thing: What are the symptoms of depression? Or, how does depression affect your feelings?
If I were asked only one of the questions, then that simplest possible interpretation is what I would assume. If the person asking wasn't satisfied with my answer they could then follow up with more questions to clarify.
If I were presented with all three questions at once, then I would assume the specific wording is important and that I'm to interpret each question differently. In that case I'd interpret question C as the simple "What are the symptoms of depression?"
Question D could be taken as asking how somebody who has depression feels about the fact that they have depression. Like, "How have you responded to learning you have depression?" So an answer might mention that it's not uncommon for somebody just diagnosed with depression to feel a degree of relief, because the diagnosis removes some of the confusion they were feeling about what was happening to them, or to feel anger that it took three different doctors to get the right medication, etc.
Question E could be interpreted in a similar way to question D, except that "would it" implies more of a hypothetical situation. So rather than asking how somebody with depression does feel now, it is asking you to describe how you think you might feel if you got depressed later.
I don't think a more in depth analysis is possible when the questions have been presented without any context.