Welcome to fanmail dtrain123!!
1. A SAE is a self addressed envelope. A SASE is a self addressed stamped envelope. A SASE is better to send because you have a better chance at getting a reply, because then the celebs won't have to pay for stamps and envelopes. With Carrie Underwood though, I don't think I have seen a recent success without a SASE, so your best bet would be to send one.
2.You don't HAVE to send a picture to be signed if you don't have one or don't want to. Carrie Underwood has her own pictures she sends out, so if you don't know what kind of paper to use, don't worry about sending one. She might not be able to sign your picture though since she is probably not home. I believe she pre-signs hers, then whoever handles her fanmail sends one back in your SASE.
But, back to the question, it doesn't really matter what kind of paper you print it out on. Some people get them printed at a store, so it looks like a real photo. Some people, like me, just print them out on normal computer paper. But probably the best to use, like dgangel said, would be high quality glossy photo paper. But any kind works fine!
Good luck, and be sure to post your feedback!!