The laptop is still non-functional, but my backup is in place and Help Desk returns!
I've received some good advice from people who have suggested that rather than assuming my troubles stem from a loose connection on my motherboard I should consider whether or not my hard drive is dying. I hadn't thought of that simply because the laptop is new, but I have to admit it's doing exactly what an old desktop computer did when it's hard drive was about to die.
If this is the case, it makes the whole thing easier to fix (though it requires the purchase of a new hard drive). Thanks to those of you who emailed in with your advice.