I have a desktop at home where I grind most of the time but I would like to get my laptop set up so I can play outside when the weather allows it. My desktop is only turned on when I'm actually using it so the laptop would have to have it's own database.

Is it possible to install HEM and PostgreSQL on both the desktop and the laptop and (automatically) sync their databases?

i don't think that this is possible.
easiest solution would be to turn on your PC and connect your laptop-HEM to the PC-database over WiFi

Yea that would be the easiest but then my PC would have to be turned on all the time which I don't like.

When HEM imports hands it deletes the HH files and creates new ones in the archive folder right?

What if I set up my laptop to also auto import the files from my desktop's archive folder (and disable archiving for those files on the laptop), and set up the desktop to do the same for the laptop's archive folder?

That way each computer should import the hands that have been played on it, after which it creates archive files for the other computer to import. After the other computer imports the archived hands they aren't archived again so there should be no infinite loop.

This should work right?