Your idea with the velcro sounds smart as long as your dog isn't too much of a mover during the night. If you used a square sleeping pad (thermarest seat??) then it would be easy to use your idea of having any side the potential opening, but maybe your dog is too big for that pad.
You may find it easier to use a slightly bigger quilt that doesn't velcro down but rather you just lay over top?
For bug protection, you could get a dog tent made like mine:
My dog tent would keep her a bit warmer than just being outside too. We use the CCF pad from my wife's backpack inside the tent (shown in that thread) for her to sleep on. I don't use any quilt for the summer time, but I can see wanting one for shoulder season trips.
If you do want a bug tent, it would be relatively easy for someone to make. Instead of attaching it to the pad, you could go Hexamid style and have it with a mesh floor so that the pad just goes inside the fully enclosed bug shelter. That would be simple to make and you wouldn't have to bother with velcro'ing the bug netting down, but you would need some kind of a zipper door.
If you had a fully enclosed bug netting tent that hangs off the tarp, then that would keep the quilt basically on top of the sleeping pad if the tent was sized to just fit the pad. That might be a little simpler and be less hassle if your dog does fidget during the night.
BTW, Bender @ Kookabay could whip you up a custom sized sleeping pad for you dog. I bet he could add the velcro too, but I think it would need to be the stick on type because sewing it on would put puncture holes in the pad. Check him out: http://www.kookabay.com/
Alternatively, you might want to use a CCF pad for simplicity. If you used really thin CCF (ie. 1/4") then you might even be able to sew the velcro to it.