here are the 5 newest minidump files( only one is in 2009, don't know why)

Shame on me.. I was in a hurry earlier today and quickly grabbed your minidumps. I saw the quick comment about only one being in 2009, but I mis read it for something different. I looked at your minidumps again, and I realize that I analyzed the ones from april of last year thinking they were from april of this year. The only one in this year is unrelated to the other errors and not related to your modem driver. It is a more generic error and signifies more of a hardware problem.
Can you verify that there are no other minidumps for 2009 in there? If you see a Blue Screen of Death a minidump should be created each time. For occasions where the screen just goes black it is probably not creating a minidump.
I'm concerned this is more of a hardware issue now as I re-read your original post. Possibly a bad connection seems how the machine crashes a few minutes after it is turned on and then does not do it anymore. Let me explain why. When you first turn the computer on after it has cooled down, the system begins to heat up. This cause metal contacts in the computer to begin to expand and will cause problems if there are loose connections. After the machine is hot and running that expansion stabilizes and there is no more "loose connection" problem. Just a guess, but it sounds logical.
Please verify there are no other minidumps for 2009. Each minidump is associated with a blue screen of death. So, they should have a date and time stamp of each occurence of your blue screen you mentioned.
Next, it sounds as if you are talented enough to get into your computer and clean it out. Thats good. I don't have time at the moment to look at the specs on your computer again (and I'm not sure the information would be there). Do you know if you have an onboard video card, or an add-on video card? An add-on video card plugs into the motherboard, where an on-board video card is part of the motherboard. If you have an add-on video card, it would correspond better with your blank screen problem possibly. Maybe the video card has a bad connection where it plugs into the motherboard. Memory is also another candidate.
You should perform a surface scan on your hard drive to be sure that it isn't having any problems. CLick start -> programs -> accessories -> right-click command prompt, click "Run As Administrator." When you are at the command prompt, type chkdsk /r and press ENTER. Answer yes to schedule a scan during your next reboot. After doing this, reboot your computer and be patient while the chkdsk scans your hard drive for file system and physical problems. Pay attention to any messages about bad clusters.
Get back with me on any new information you have about the frequency of your problem, and any other blue screen information you may have.
Ignore the modem driver suggestion, because that was a problem back in April of 2008. I apologize for not paying closer attention. I assumed it was april 2009.