If Apple is willing to send you a new Mac then it sounds like they are conceding that yours has a hardware issue.
Man, some good advice here, some horrendous. The level of misinformation really is scary. This kind of troubleshooting is one of those things that becomes much easier the more comfortable you get with OSX. Knowing Unix is a HUGE advantage because you'll already know how to navigate logs, permissions, startup folders, etc. from terminal.
I wish MacRumors had a 'reputation' feature or some other scoring method we could use to evaluate the legitimacy of advice/users. I love OSX in major part because I love and know Unix. UNIX is fast, powerful, and robust- that said, it will let you eff yourself if you tell it you want to. Or, I guess it's more that it will happily eff itself if you tell it you want it too and say please(= sudo)!!! =) Be very very very careful following step by step solutions if you don't know what the steps are doing.
My opinion, (I liked IJ and applespider), never ever ever reinstall the OS as a step in the troubleshooting process. This is terrible advice and there's no argument for a "agree to disagree" scenario. You effectively destroy your greatest asset in reaching a solution: the ability to reproduce the problem. Reinstalling is not a fix, it's a total game of chance. To reinstall without knowing what you're trying to accomplish is just desperation. I really think you'd be hard pressed to find many competent admins/users that would "agree to disagree" like suggested.
One of the most important things to remember anytime you're mucking through problems like this. it's not worth wasting too much time. You have to keep weighing the likelihood of getting it working, the cost, the benefits. Keep a mental record (or on paper if you're organized) of how long you're spending. If it takes you a week to fix it- you should have sent it back and not lost the sleep (and gain some sleep from not having your mac
.... ) I consider myself capable of fixing many things but I can think of countless times I should have followed my own advice. You can waste the time, lose the sleep, lose data, lose money, and never reach a solution and on top of that you might not even really learn anything from the hacking. The more you know, the harder it is to concede but it is the sensible thing in many situations. We pay for the applecare, might as well put it to work when you can.
If I had your symptoms I hope that I would test the ram immediately. It's the first thing that came to mind when I read the OP. In reality, I don't want to believe that my ram has died and I exhaust every avenue while booted before I dig out the OSX dvd wherever it's hiding.
Here are a few things off the top of my head that I do if I am troubleshooting:
a. repair permissions
b. verify/repair disk (from DISK UTILITY)
c. Remove any unnecessary devices.
d. Sift through login items, startup items. Make sure nothing weird is going on. Disable them, add them back slowly until panic.
1. Login as another user, try to reproduce.
2. Login in safe mode, try to reproduce.
3. Examine the logs for each kernel panic, see how they compare, look for common thread.
4. Test the ram.
