"stand up" is not a good verb to use with "contract" but is a good verb to use with "claim". "claim" and "contract" are not synonymous or even in the same word "family". You use "stand up (to scrutiny)" when considering the merits of particular claims or allegations, and in this context "to be valid" is a similar way of expressing this.. e.g. "His claim that West Ham United is the best football team in England is not valid / does not stand up (because the team is in 9th position in the league)" If a contract is not valid, then you probably mean that the contract has no effect because of some legal mistakes. Lawyers would probably not use the word "valid" but it's good enough for general usage.

That would depend on the context. 'doesn't stand up' is much more informal and less specific than 'isn't valid'.