HP Pavillion notebook reboots continously except in Safe Mode

This is a family member's computer and she asked me to help her fix it. It will only stay booted in Safe Mode. When I start the computer, it boots up, stays booted for a few mintues, then the screen goes black and it reboots. This keeps happening unless it is booted in Safe Mode.

History:
Purchased with Vista; I clean installed Windows 7 Home Premium (I know it's legit, bought it for her myself) when Windows 7 first came out.
When I turned it on, it rebooted so I started it in Safe mode and have tried/found the following problems:
· It had 4 adware programs and one trojan horse; removed those with Malwarebytes. Rescanned with Malwarebytes and MS Security Essentials, came out clean
· Device manager showed two problems - bad drivers for co-processor and bad drivers for Graphics card. installed those drivers and the Device Manager did not show anything wrong. Problem keeps happening.
· Opened BIOS and ran both Hard Drive and Memory Diagnostics; no problems found, Did not make any BIOS changes (I have little experience with BIOS)
· Ran diagnostics with Windows 7 disk; including Repair, Memory Diagnostics; no problems found
· Graphics card is NVIDIA GeFroce 7150M/nforce 630M; reinstalled drivers, didn't help anything
· When the screen saver goes on, there is an error message, "The screen saver can't run because it requires a newer video card or one that's compatable with Direct 3D".

This is an old computer, well out of warranty.

Do I... still possibly have an embedded virus? What removal tool should I get?
Is it worth replacing the graphics card - if that's possible on an old laptop?
Should I try a clean install of Windows 7? Seems like if I have a hardware problem, this won't help.

Thank you.

Sounds like it might be time for a reinstall to refresh the OS and overcome any lingering infection. Follow these same steps for Clean Reinstall - Factory OEM Windows 7 particularly when it comes to drivers and Updates.

Otherwise work through these Troubleshooting Steps for Windows 7 including the hardware tests.

I can work for about 2 minutes before it reboots. I was able to install the drivers through safe mode. HP has a program that checks for bad drivers but it won't run in Safe Mode, and the computer does not stay on long enough to run it.

I'll try the clean install suggestion. Our "victim" has very little data to save - just some address book entries, a few files and almost no other software installed, except for Works, which she doesn't use, and Adobe Reader.

I'm sure part of the adware stuff that Malwarebytes found was due to her downloading "free" screensavers.

The laptop is about 5 years old; I'm not a fan of HP but this one has been decent and so far,and has not overheated. That's probably because she hardly uses it.

I'll post what happens.

Quick Question - would I download the LAN driver from HP's site for this laptop and save it to a flash drive, then load it after Windows 7 is installed?

I'll look at Greg's link, I dowloaded it and saved it and clean out the vents.

My Windows 7 version is not an OEM - I bought it at Office Depot. The laptop came with Vista and HP's "wonderful" self service center which did nothing but cause problems and slow down the computer. Comcast's free Norton, which is a memory hog, didn't do help matters.

Did a clean install of Windows 7, worked like a charm for the past few years until now - when our "victim" downloaded "free" stuff. Yikes.

Thank you everyone, for your help.

Turns out the laptop had a Bot virus which I removed from Kasperky's virus removal tool. After that, everything has been working fine.