They said they would look at it...
Does that mean you didn't actually go there in person yet? Frankly, I'd be looking to get a replacement unit ASAP. It seems like you have a hardware problem.
One item that recently occurred to me is that I've got 16GB of RAM from OWC. I've not tried dropping to a single DIMM, nor have I tried going back to the original 4GB of Apple RAM.
It also SEEMED like the EFI Update complained during the install, but it flashed by too quickly for me to be sure. The Boot ROM version is consistent with a valid EFI update, and when I try to manually install it, it says it's already installed.
Ah-ha. Now maybe you're getting somewhere. It's entirely possible to have a bad ram chip. I forget what it's called, but I had a program I used when I upgraded my PowerMac to 1.5GB and its CPU to 1.8GHz to test the memory over an extended period of time from a cold boot. I was having problems with sound disappearing in OS9 (appears to be related to the CPU upgrade) and I had to make sure it wasn't memory related. Not that it mattered, the company wouldn't do anything about it (Newer Technology G4 upgrade) so I pretty much gave up any hope of playing old OS9 games at that point (productivity that didn't need sound was fine, etc., though, but I would get sound about 1/3 times on the first boot to OS9 after a reflash, but I wasn't going to do that every time I wanted sound in OS9 and it would stop working again on the next boot).
Anyway, my point is it would be a good idea to try swapping back to the original memory and see if it makes any difference or at least test the new memory extensively to be sure there's no issue with it (frankly, it'd be easier to just swap back; the new Mini is very easy to open).