My bike is whole again!
Thanks to everyone for all of your good advice. I've learned a lot about tires and tubes during this little ordeal. As it turns out, I never had a tire problem-- I had a tube issue. My tires are old and "wrinkled," but they're totally sound; I looked them all over, inside and out. My poor old rim tape dry-rotted and one (or more) of the spokes wore through and ripped my original tube. Then the replacement tube was either defective or I pinched it when I was trying to get it back in the tire. At least, I didn't do something horrible to my bike during repair attempt #1 because, Earl, my church's drummer came by. He used to work as a bike mechanic and he put my wheel back on for me. After watching him, I felt confident in undertaking repair attempt #2, but I didn't put everything back quite right (as Thom and Aaron sharply observed in the previous post). Now everything is in its right place and I can ride my bike.
Thanks again, everybody!