If it was me the first thing that i would do is to remove the remote starting system, unless it is factory. which i am sure it probably isn't because it is a 93. I would guess there is probably something wrong with the remote starter, or it is installed wrong. It sounds like for some reason the remote starter is leaving the ignition in the on position and probably draining the battery sometimes. That may also be why it won't start with the key. If you really want to have a remote starter, i would take it someplace that installs them all the time, and put in a brand name remote starter in like viper.
Thanks and good luck! i hope this helped
ASE Certified Technician
5 Years Experiance 2 Year gm dealership experiance
Thats great to hear!! it makes me happy to help you!
Thanks again!!