1) Is it rude to not say 'Thank you' when someone says 'Bless you', or acknowledge it in some other way?
I've been involved in many discussions where saying 'Bless you' is actually considered rude and presumptuous because it's religious in nature. In general, where I live, most people will say 'Bless you' and that is considered the most polite thing to do - to acknowledge that someone else had a poor health moment (a cold or an allergy) and that you are wishing them good health. But from the discussions I've had with others, IRL and on forums, not everyone takes a 'Bless you' as an acknowledgement of what they did and that you wish them good health. Some people truly believe that you are making assumptions about their religious beliefs and the state of their souls and they dislike being told that. Those folks are generally ok with 'Gesundheit' based on the things I've read.
2) Is it rude to say 'Bless you' to everyone in a group except for one person? Does it make a difference if that person has never acknowledged a 'Bless you'? (Note: There are other people besides me that still 'Bless you' Ann, but I don't know if she would notice even if there weren't.)
Generally, yes. But if she doesn't seem to appreciate it, then are you truly saying it for her benefit or for your own, so that you feel good about being polite and thoughtful? Maybe she doesn't notice, so you shouldn't expend the effort. Maybe she secretly dislikes it and hopes that you'll stop.
3) Is it rude to never say 'Bless you', 'Gesundheit', or sneeze-related comment of your culture when someone sneezes?
I think it depends on where you live. I live in a roughly cosmopolitan area with various religious and cultural backgrounds so people generally say 'Bless you.' I've lived in other places where it's more expected, like said about 95% of the time, and other places where it's said about 60% of the time.
4) Do the answers to the above questions vary based on whether the person is close (friend, family member), not-so-close (co-worker, acquaintance) or complete stranger?
I'd say yes. If you're good friends with someone, you might know their religious beliefs or if they notice people saying that to them. Someone who is seated near you but is not a close friend, you don't know if they like it or not.
In your situation, I've actually said to my coworker "I tend to automatically say 'Bless you' to people, but I don't want to do it if it's annoying you rather than coming across as a social nicety. What do you think?" And then going forward, I try to do what interrupts them and annoys them least.