1) When the PC is closed down but still on at the socket it sometimes boots itself up.
This can be caused by BIOS settings, which allow various events (such as the 'phone ringing) to wake up the system. Go into the BIOS and disable all *wake-up* events.
2) When the PC is on it will sometimes dial up the ISP all by itself.
Some programs can cause this - even legitimate ones such as AV programs trying to do an update. However some malware can cause it too. So check for viruses with an up-to-date AV program, and check for malware (rogue diallers in particular) with Adaware and Spybot.