Well for me it still sounds like a car accident and the Present tense seems to be just a little mistake. Tell me please, what do you exactly want to say?
You're right that it can also mean that... But when I hear "kolizja" I automatically think about a car accident and I can't help it! Maybe you should say "mam kolizję spotkań" ? Anyway it sounds a little odd for me...
Same here. I'd say then, nakładają mi się spotkania. If you pushed me in a corner and twisted my arm to force me into using something related to kolizja, I'd use spotkania mi ze sobą kolidują. The Polish word kolizja has too many car-crash overtones to use it like that by me.
I don't think there's somethng palatable in Polish where you could use a noun (as a direct translation of conflict), at least I can't come up with anything, perhaps someone ese can. I'd simply use some verbal translation as it works best to me.
Well on the other hand:
Nie mogę przyjść na spotkanie, mam inne/drugie w tym samym czasie.
but that's a different formulation from the English one.