Blue screen of death!

In my experience if a computer starts to boot fitfully then the hardware
problem is usually the power supply or the hard drive.
Much gets blamed on motherboards but a functioning mobo rarely goes south
unless some defective add-on or power spike caused it.
What you describe is less often a ram problem but you should be able to run
the Microsoft Diagnostic which very rapidly will identify bad RAM.
If the computer boots at all it is not likely the BIOS. BIOS can be
corrupted by massively dysfunctional ram sticks (I recently experienced this
with brand new bad RAM, thank you NewEgg) but that's about it, barring
massive malware infestation which sounds unlikely. If your computer is
unstable do not attempt to reflash the BIOS as a failed flash will render
the mobo functionally dead. Only the manufacturer can reflash it at that
point.
RAM that is bad out of the box is all too common; good RAM seldom goes bad.
Manufacturers, and I single out Mushkin, rarely honor their lifetime RAM
warranties.
If you can boot the computer at all BACKUP YOUR DATA NOW. If your hard drive
is dying this may be your last chance.
If you can boot the computer delete and reinstall the video driver: a
corrupt video driver can cause this, but BACK UP YOUR DATA first.
Impending hard drive failure, and in my sad experience there is a high
failure rate in brand new large hard drives, cause random lock-ups,
difficulty booting and can suddenly cease altogether. You may or may not
hear the click of death prior to hard drive failure. High quality power
supplies tend to be more reliable than hard drives but they can fail also.