Zeke is correct. You can say "how are you?" For a shorter version or "how are you feeling" if you specifically want to know their health. Same concept with "are you feeling well?" Usually implies you want to know their health condition. Though, most say "are you feeling okay/alright?" There are many ways to express this phrase but those are still correct. The only difference is that other expressions like I used above get into specifics.

I believe both are suitable. Since the structure of the phrases are correct and they both ask of the same things, both could be used. However I think ''do you feel well'' sounds more polished compared to are you feeling well.