No, spelled the same: https://en.wiktionary.org/wiki/noi#Romanian
A/the standard meaning of "homonym" I think is two "words" spelled the same but with unrelated meanings and possibly different pronunciation. Confusingly, one can find "homonym" defined as "homophone," which I think is the sort of situation you probably had in mind.
yes: noi means we; but noi means new only for plural nouns, like in: new shoes = pantofi noi; new girls = fete noi (and 'noi' should stay after the noun). if you put it in front of the noun... it is more complicated, noii pantofi = the new shoes; noile fete - the new girls. in this last case it emphasize the fact they are new/fresh.
Yes. For first conjugation like mânca, add -ăm, for second and third conjugation like avea or scrie, add -em, and for fourth conjugation like dormi, add -im. It's sort of like Spanish with -ar, -er and -ir verbs.
a mânca (to eat / comer)
eu mănânc / tu mănânci / el, ea mănâncă
noi mâncam / voi mâncați / ei, ele mănâncă
a avea (to have / tener)
eu am / tu ai / el, ea are
noi avem / voi aveți / ei, ele au
a scrie (to write / escribir)
eu scriu / tu scrii / el, ea scrie
noi scriem / voi scrieți / ei, ele scriu
a dormi (to sleep / dormir)
eu dorm / tu dormi / el, ea doarme
noi dormim / voi dormiți / ei, ele dorm