Contractions of the type that have not in them should only have one apostrophe. "I would not have" is "I wouldn'tve" not "I wouldn't've."

Reza, how's the use of ain't taught there? Avoid or use?

I've never heard the possesion deal with contracting have. That's not a rule I ever heard of, but in your example sentence I would say the word have and not contract it. I think it depends on what is being talked about or possessed. Can notions be possessed? Such as, "I've a notion to do something about that?" I've noticed the over use of the word got, especially when it follows I've. It seems to me if you wouldn't say, "I have got it", then you shouldn't say, "I've got it."