Pretty simple, open the remote...most likely just snaps apart, replace with same part number battery
Possible weak battery in the remote.
You might need to programming the remote. Their should be instructions in the users manual. You're remote could also be need new batteries.
Check instructions in the users manual.
Sounds like the contacts are still touching some how. Take the keyless apart and spray electronics contact type spray on the contacts and the points where the battery connects to see if this will help. Are you using a brand battery or a cheap-o?
I had the same problem. Google "keyless entry remote 2003 Jeep Grand Cherokee".
Im not sure if you're talking about remote starters or alarms, but the alarm light did pop on when I changed the battery in my 95 Jeep GC Laredo. All I had to do was lock and unlock the doors a few times to reset the system.
It is easy to do with online instructions. Just google "2004 Jeep remote programming"
You have to program it with a DRB (Scan tool).
centre top dashboard (red dome)
The dealer
dealer has to do it with factory scan tool
If the key will not work in the door or the liftgate, you will need to call a locksmith or break a window.