There are virtually no words, in any form, that have -żi ending in Polish. I don't know much about any rules on endings in inflection, but it will have to be -ży and not -żi if the stem ends with a ż.
I know that some people disagree with this decision, but we do not accept "laying" even if some native speakers use it. Just like we do not accept things that can be said by Polish native speakers but are considered grammatically wrong.
When does 'na' take genitive as here, and when does it take instrumental?
Never and... never ;)
It's Locative here, it simply denotes location (on the bed). (not literally all prepositions denoting location take Locative, but most of them).
The alternative is Accusative, mostly used with movement.
Skaczę na łóżku = I am jumping on the bed = Locative.
Skaczę na łóżko = I am jumping... to? the bed (from the floor on the bed) = Accusative.