Where do I update my BIOS?  

Where did you get your copy of windows from?

I don't usually recommend this, but I'd suggest taking your computer to a repair shop and having them look at it, or calling HP if it is still under warranty. Updating the BIOS likely won't solve any of these issues, and based on your questioning, I'd be worried that you'd do some permanent damage to the system.

The exact model number is on the bottom of the laptop, but you also should know this from your documentation from the purchase of the laptop.

If the system doesn't power on right away or each time you try, that is usually a sign of a failing motherboard.

How old is this laptop? What kind of shape is the battery in? If the battery is dead or quits in a short period of time, try booting up the laptop with it removed.

The missing co-processor is a known issue solved by installing the full chipset download from the HP Support Downloads webpage for your model computer. Download it to desktop, extract the files, then rightclick the .exe file to Run As Administrator.

If this isn't sufficient, use the scanner on the Nvidia Drivers download webpage which will tell you what chipset you have and provide the latest driver.

Work through these troubleshooting steps to make sure you have everything covered.