Cadillac Repair: Numerous and Random Electrical Issues - 98 Deville, mileage info, faub

Question
We recently purchased a used 1998 Deville and have had random issues with it.  We replaced the battery, and changed the factory installed radio.  We also replaced the battery in the RKE key faub, and had all cables cleaned and tightened.  However - occasionally:
1. The car stalls when stopped at intersections, but restarts just fine.  No warning lights.
2. The RKE works inconsistently.  If I open with a "real" key, the alarm goes off as soon as I start the car.
3. All diagnostics reset - forgets time, trip odomoter, mileage info, and radio presets.
4. I put the car in park, and the car just dies - again, restarts just fine.

I can't help but think these might all be related, but I can't get any of this to happen consistently.  Any ideas?

Answer
If you are mechanicaly inclined or know of someone who is, You will need a DIGITAL MULTIMETER, Simple electrical schematics, Test procedures and picture locations of the components within the system that is causing you problems of which can't be done on this website which only allows one picture. I do not have a website so you will have to e-mail: onlineautohelp@msn.com in order to obtain the information.