This doesn't look great - let's remember we started with an integrated graphics chipset on a motherboard that was throwing blue screens. That's still the backbone of the machine even though it has a new graphics card. You'd not opened the case between packing it in the car and finding it didn't start (or even POST) so we've got to think something moved - either because it wasn't secured or because it's cracked (PCB or solder joint). Movement can cause one of two scenarios :
1) A contact that should be there is broken
&
2) A contact has been made that shouldn't have
You've already checked that everythnig is securely connected so is there anythnig that has moved and may be making a connection it shouldn't (the most common is the motherboard touching the case)?
One other thing I'd try before giving up on it would be to jump reset the BIOS incase it's been messed with trying to detect something that had moved but with no POST beeps at all this is only an outside chance.