Do I need to purchase a sit for my 2 year old for Disney on ice, if she is just going to sit on my lap?

You're not going to verify your child's age if they're just sitting in your lap I'd say you're probably okay but that's just my opinion hope this helps

2-12 requires a ticket, but they don't check an ID.

Well no I don't think so because with her being that young it's free I believe

Sometimes you do