Why is this są and not jest?
"urodziny" is plural always. Just a quirk of the language.
urodziny it is plural always, jest is for singular : to jest kobieta , to są kobiety
W lipcu są moje urodziny?
Well, that means that you've been talking about July (and so it happens that you were born in July), not talking about your birthday (which is in July).
But... okay, added.
What gender is urodziny? Wiktionary does not specify, so I was wondering if "Mói urodziny są w lipcu" would also be correct, or if there is a rule for this word?
It's 'not masculine-personal plural'. This word has no singular form.
And "mói" is not a word, I guess you meant "moi", but it's the wrong plural.
Why is "Mam moje urodziny w lipcu" not right?
You already have the conjugated verb, so "moje" is redundant.
I think I tried it without moje too, but rejected the shorter 'Mam urodziny w lipcu' as well.
OK, added "Mam urodziny w lipcu", but adding "moje" is indeed pretty weird, of course it's your birthday.
dlaczego nie: w lipcu mam urodziny?
Why not, added.