Not a very stable computer you have there... I had XP SP2 running just under 8 weeks straight without a reboot ... and I only rebooted so I could swap out a PSU. XP is quite stable even if you run a lot of apps (install/uninstall etc). If you have problems, it is often some form of spyware/adware/trojan/virus/etc or else you made some setting change which was not a good idea.
Regarding the question of "is it a mIRC problem or a Windows problem?" ... it is a Windows problem, but mIRC should still redisplay itself. From what I've seen only perhaps 10-15% of apps redisplay in the tray if you kill explorer (on purpose or from a crash).
