Saying "I got" is wrong in this context. You may have heard people saying it, but it's still grammatically incorrect. If you say "I got" you may sound uneducated. You can either say "I have got nothing to do" (shortened form: "I've got nothing to do") or you can say "I have nothing to do". You should only say "I got" when you are using the verb "to get" in the past tense. "I got" does NOT mean "I have". People who use "I got" to mean "I have" are usually rather uneducated people.

You got nothing better to ask?

Your first sentence should read "I've got nothing to do." instead of "I got nothing to do." the difference between the sentence "I've got nothing to do."( "I have got nothing to do.") and "I have nothing to do." is that when you place "got" after "have" you are adding emphasis to the fact that you have nothing to do. It's a weird semantic. In my opinion one should always use "I have nothing to do." because that sounds a lot more literate and formal.