Not really, but it is something to keep an eye on, just don't let the zipper catch on the hammock. I've used a sleeping bag for a long time.
Underquilt or pad is really a matter of personal preference. If you are just starting out I'd suggest finding a wide pad for now because they are generally cheaper and you can use it in tents if needed. It may be a bit annoying to get set up in the hammock, but should work well.
You can research both in the bottom insulation section of the forum. What kind of pad/underquilt also depends a lot on what temperatures you will be sleeping in and how much weight you are able to carry.
I personally used a 20 degree hammockgear phoenix underquilt, it has worked well with my hennessy and has kept me warm in a lot of temperatures. The down quilts can get expensive though so you should do some research and figure out what your needs and preferences are. A closed cell pad will work in the meantime. Good luck!