Yes, with wonderful results. Since fabric is so thick don't hoop...use sticky back, pin or baste your fabric in place.

I have embroidered on quilts, without backing. I just hooped the quilt. It's tricky, and I broke a few needles in the process, but if you're careful, it can be done.

I have with great results--I tried to find in projects but couldn't, but did find another sample--she used prequilted from Joann's also.

I have on the pre quilted fabric from Walmart

I haven't tried it yet but I have some quilted fabric I bought to make oven mitts. One of these days......

I just bet that they would turn out wonderful. Here are some projects done on pre-quilted fabric:

Yes I use prequilted fabric & placemats for my food carriers. I especially like not having to use stabilizer with them. If it's heavy fabric then I will hoop stabilizer & baste the fabric to the stabilizer rather than try to hoop. You should have no problem making the bag, just embroidery the design before you make the bag. It's easier that way. *

I made Christmas stocking from pre-quilted from JoAnn's, they turned out really well....this year I hope to do a Christmas tree skirt...I'm sure you will have great results.