1) Michael, do your coat up!
2) Michael, zip your coat up!
Do they mean the same? or Had we better use "to do up" when there is no zip ( for example, there are some bottons) and "to zip up" where there is a zip to close.
You can omit "up" with zip.
Zip your coat! And put on your gloves! It's cold! (And even if your children have blue lips, they will say "We're not cold!")
I think Americans are less likely to say "Do up" -- if the coat had buttons, we'd probably say either "Button your coat!" or "Fasten your coat." I think. My kids have always had zippers and I wouldn't tell anyone else what to do.
I'm not a teacher, but I write for a living. Please don't ask me about 2nd conditionals, but I'm a safe bet for what reads well in (American) English.