[/quote]That "cold boot" is exactly what I did. I don't want to have to do it every time I want to reboot.[/quote]
I agree absolutely, it's a quick fix, not a permanent solution. HP recommends it when the machine fails to start Windows recovery properly. So cold starts are fine, system reboots are the problem.
Another thing I noticed is your boot sequence seems out of order when compared to one my laptops. On my HP Envy DV7, with Windows 8, I see the message to press the Escape Key first, then I get the HP logo splash screen, then I get the login screen. If your machine is hanging at "press esc to enter setup", it seems your problem is occuring prior to the OS actually loading, possibly during power on self test (POST), or maybe when BIOS is passing control over to the bootloader in the MBR. Since your machine is new, I would think it has the most current BIOS version installed, but it may not hurt to check. It might also help to check if any BIOS settings have been changed from the default settings. Usually on HP laptops, pressing F9 will restore all setings to their defaults, then pressing F10 will save the settings and exit. Please read the instructions on the top or bottom of the BIOS menu screen to verify that F9 and F10 perform the same functions I described before doing this. Again, this is a strange situation you have here, but if what you described is accurate, I'm tending to believe the problem is BIOS related and not related to either Windows or Mint.