AYou want to quilt it's fresh, right after you've covered it. It's going to be a bit tricky because the dummy cake is light and will move around.
Also it's a lot easier for me to cover the dummy with shortening instead of water. The water makes my fondant kind of gummy and I don't like that
Hello pamalbake!!! I just went through this last week. I got a lot of help from very generous ladies of CC.Pls take some time to read this post http://cakecentral.com/t/777095/cake-dumies-fake-i-need-help
I hope it helps you, it sure was lots ofr help for me.
Good luck!
Ciao
P.S. To answer your question, I quilted right after covering dummy cake with fondant.
good morning! As I have read, yes you can reuse it. I imagine that you take the fondant out and would just clean it with a paper towel, I would not use water, the foam might absorb it. I never asked about it because cake dummies were not mine, my friend bought it. If you have more questions about it, I would Google for info " how to clean cake dummies"
HTH
CIAO!
It also depends on how many times the dummy was used, because they sometimes tend to wear down. If you need more, I buy mine in bulk from buyfoamblocks.com they are different quantities you can get. I go through a lot of them so its nice to have some extras! 
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