As has been described a number of times, you can't program out VATS on a 4th generation f-body. VATS is part of the BCM which is not programmable. It does two things to protect the car from being stolen - it disables the fuel enable signal which the PCM relies on to start the fuel pump; and it disables the ground wire of the starter relay preventing the starter from turning. You can program the PCM to ignore the lack of a fuel enable signal, but that still leaves the starter relay problem. You can rewire the starter relay to use a different ground, but that still leaves the security light in the instrument cluster. No amount of programming will fix that although a piece of electrical tape can hide it.
No, the best thing to do is a VATS bypass using a few inexpensive resistors. It will solve all of these problems. There are numerous threads on this site about how to do a VATS bypass.
As far as your fob goes, it sounds like it may be on its last legs. You could try re-matching it to the system, but you're probably better off getting a new one (even better, get two so you have a spare).