They are spelled correctly. And here's a comparison. - 진짜 - very informal, - 정말 - relatively informal. - 사실이야, 사실이에요 - more formal. 사실 (fact) can't be used as an adverb or exclamation like 정말 and 진짜.

both are correct, and if you put 요(yo) at the back, both are ok in formal situations too. but you can't use it all the time, that would be weird. if it isnt a big thing, just say 아 그래요? (ah, grayo?)