2001 SC2 alarm on/off/wacky

Bought this car a couple of months ago. Runs nice but the alarm is
kind of driving me crazy. First problem was with the first very hot
days of Spring here. The alarm would just go off for no reason. I
took it to Saturn dealer. They said it seemed fine and they couldn't
fix it because the problem wasn't manifesting itself at that moment.
Went home and took the horn fuse out which solved the problem except
for not having a horn. After a couple of near misses with crazy
drivers around here, I put the horn fuse back in. Now, I can lock the
doors with the remote but when I go to unlock them with the remote,
nothing happens. I have to use the key which starts the horn honking
until the key is in the ignition, totally ****ing off my neighbors
again. They are not liking my new (to me) car. I tried the "press
any button three times Automatic Synchronization" trick and that
didn't do anything.
Any suggestions?
Thanks,
Tom