Cookie Cutters work great on the fondant. I prefer to use fondant on my cookies because it comes out the same size as the cookie.
It will definitely dry if you leave it out. In order to speed up the process, add a little tylose to the fondant.
Good luck with your fondant endeavors!
I use the readymade fonant, color, and roll it out using different cookie cutters to make shapes. Let it dry on parchment or wax paper completely then use to decorate cake. They do harden great. I even place the leftover one in a plastic container to use in the event I have a quick order to do.
If you rub a little crisco around the edge of the cutter you will get a cleaner cut edge on them. 
When I do this I use rasberry preserves or orange preserves to glue them on the cookies.
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