I don't think it's rude. This is an office setting, not a dining room table or a restaurant where people are eating and looking at you. Practicality is called for and, realistically, people can't get up and leave their desks.
As far as hand washing and sanitizers--right now my hands are raw and cracked from washing them often. I use a heavy hand cream that helps but does not cure it. The hand sanitizer REALLY does a number on my hands. So I only use that in a emergency. I'm lucky in that I have rarely had issues where I produce a lot of mucous. So a box of tissues might sit in my office an entire year and not get used. Only the most severe colds have caused me to mess up my hands when blowing.
That said, if I do blow my nose I keep my hands to ONLY my stuff. When I have to get out in the public, I make a quick trip to the rest room and wash them and then come back and wipe my keyboard and mouse down. That way, I only contaminate myself for a few hours and then clean up before I use the copier or microwave. No one else uses my stuff.