I thought I would give an update to my situation now that it's been resolved. Initially, the computer would get stuck on the gray screen and give the three beeps randomly. Sometimes it would boot up like normal, and other times it wouldn't. It seemed like more often than not, it would boot up fine when it was plugged into the wall instead of just running on battery power. This problem cropped up out of nowhere, because the computer had not been dropped or banged or jolted at all.
I took it to my local Apple store, where it failed to demonstrate my error for the Genius (of course lol). He did confirm that the symptoms I was describing indicated problems with the RAM or RAM slots, but since the error wasn't occuring, he really couldn't help me. He did re-seat the RAM cards for me just to be sure, and sent me on my way telling me to download all the latest software updates, and to bring it back if the error occurred again.
As soon as I get home, the error happens! I made an appointment to go back to the Apple store the next day, where it did actually demonstrate the error for the Genius. She said that I would need to check it in to be repaired and she disappeared to the back to quote me a repair time. When she came back, she instead offered to exchange the computer for me, and since I had no important data on the computer, and I backed up what I did have, I accepted. Everything has been working fine ever since.
