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When it will not start, what exactly happens?
The other problem that I had prior to this is that the remote did not work and has not worked, then the radio system went off for sometime andf started up again on its own. Just yesterday we installed the new batery but still having a problem with starting the vehicle it worked well while we were in town but then it stopped and started back again and now has stopped completely..
Since you are having the problem with the remote I would reccomend letting the dealer reprogram it as these may be related. If that does not solve it, some of these have issues with the ignition switch acting up.
But I think there is a good chance just the reprogramming by the dealer is all that is needed.