Idioms are tough, because if we don't know them, we feel helpless! I disagree with Ron on this one: although 'proof of' is definitely more common, I think that 'proof for' can be idiomatic (
https://en.oxforddictionaries.com/definition/proof).
In any case, look for another reason to eliminate answer E. The meaning is key here: have a think about it.
As for your sentences, I'd say that they're okay, but you're not really idioms of the word proof. If we say 'proof of the crime' or 'proof for the existence of God', then
the crime or
the existence of God are the things being proved.
I want proof for medical reasons. Presumably you want proof of something (such as a condition), although the meaning is a bit unclear.
I collect evidence as proof for my upcoming lawsuit. Here you're not trying to prove a lawsuit, you need proof (of something like your innocence) to use in your lawsuit.
I mean, we can think of lots of other prepositions with proof in some situations: Sherlock Holmes looked for proof in the cupboard, or under the bed, etc. But 'proof in' and 'proof under' aren't real idioms, because we're not proving the cupboard or the bed.