Acetone will attack vinyl gloves and probably latex rubber as well. It tends to dry out your skin with repeated contact but otherwise isn't all that hazardous. The main risk is fire so make sure you aren't close to any source of heat. If you browse older posts on the forum you will find lots of discussion about using acetone.
I'd just pour the acetone directly from the can into your glass containers and pour the used stuff into a waste glass when you're done. You can just leave the open waste container outside and the acetone will evaporate quickly.
After your final rinse just pour off as much acetone as you can, then gently tip the coin onto a soft paper towel. It will be dry and OK to handle in just a few seconds.
Oh, and make sure you use pure acetone. Nail polish remover (for example) is acetone with a bunch of other chemicals and would be terrible for your coins.