I really don't know anything about ghost shrimp, so I'm afraid that I can't help you with your second question.
However, Bettas are supposed to get along quite amicably with just about anything other than more Bettas, so I don't think that's the issue.
As for water changing, around here we dump the whole thing and put the fishy in a smaller container while we clean the fishbowl, but we don't have filters for our Bettas, so you may be able to get away with a 50%. I'd say try it and see how it works for you. If you find that the 50% isn't keeping the water clean enough, then do a full clean.