can i use i am not anybody's wife??
Should be able to... I just put in "I am not anyone's wife" and it was accepted. The words "anyone" and "anybody" are interchangeable.
Is it required to use double negation in the sentences like this? Can't I just say "Jsem ničí manžel(ka)"?
It is better. "Jsem ničí manžel(ka)." is quite unnatural. It strongly emphasises the word "ničí", but everybody would understand it.