Blue Screen (Win7HomePrem64bit) on HP TouchSmart 300-1223 (All-in-one)

OK I have been having problems trying to get a client's computer fixed. It is an HP TouchSmart 300-1223 which is an all-in-one. First it was viruses. I got rid of 54 of them. Then the hard drive was bad. Just swapped it out and put a brand new install of Windows7 64-bit Home Premium. Did all of the updates. Official drivers. I even put a 4 gig stick of RAM in there instead of the 2 gig stick that ended up in there somehow. RAM was tested and working BTW. Ran like a rocket ship compared to before! Then I was testing to make sure everything was good to go so I could give this guy his computer back and it would be out of my life. Got a bluescreen with the Stop code 0x0000001E so I turned off the computer, unplugged the USB keyboard and mouse, and tried to turn it back on. Fans ran and everything but the screen was not on. I did the same turn everything off, unplug everything procedure as previously mentioned only I removed the stick of RAM, turned it on, and got the beeps saying it had no RAM. I re-seated the stick of RAM, plugged everything back in, and BAM! The screen had color again. No problems with it for a day and a half running it constantly (videos playing, surfing) so I felt confident giving the client his computer back. That was two days ago and he just told me he got the same exact error I just mentioned to you guys. He is going to bring it by again tomorrow morning. Would you guys please help me out in getting this issue resolved so that I can finally get it out of my life? This thing has been at my house on-and-off for a month now. Any help is MUCH and GREATLY appreciated

(RAM is good, Hard drive is brand new, brand new installation of Windows 7 64-bit Home Premium and has been authenticated, did all possible updates) (I have no idea how to tell but I kind of think this is a video driver or power supply issue. I installed the Windows7 64-bit video driver from HP's site just in case anyone was curious and thinks that could be the issue)

I already have Security Essentials installed and up-to-date. System has been scanned before and after this Stop code 0x0000001E. This is a CLEAN install of Windows so I can assure you this is certainly not a virus/malware issue. Any help is still wanted though :)

Attached are dump files and BSOD information taken from this computer -

I have been using the wireless connection. What is odd about this? I have also uninstalled and reinstalled Security Essentials along with installing Malwarebytes Free and performed scans as of 30 minutes ago. Shortly after the scans the screen flashed red then blue then green then black and locked up. It wouldn't turn back on until I unplugged everything including removing the RAM again. Then I turned it back off, put the RAM back in, and got a color screen. After that I went to HP for the AMD SATA driver and ATI graphics driver. After I installed and restarted with the drivers I got an error related to ATI Catalyst Install Manager every boot saying a service (kdbsync.exe) stopped. I uninstalled that component (AMD APP SDK Runtime) from Catalyst Install Manager and rebooted. Still not sure what is going on but I have done a good bit and it is on now. But then again it was before. I just haven't pin pointed the exact problem yet...