As a general rule, I wouldn’t game on the same system that runs a NAS. You won’t want to be taking your NAS offline in order to mess with the drivers / pass through settings on gaming VMs. Also, gaming on a remote machine isn’t great, even with fiber connections at both locations.

Buy a gaming/CAD PC for you and your gf and make a NAS out of a raspberry pi with a USB drive while you learn the ropes.

You should be able to find a used consumer tower with something better than an i3 10100 in your budget if you let go of ECC.

Don’t worry too much about making wrong choices. You will eventually find yourself wanting to build a whole system from scratch no matter what choices you make now.

Do you already have experience with docker and some of the software you want to run? If not, I’d recommend starting there on whatever the cheapest system you can find. Unraid and TrueNAS make a lot of choices for you and can provide a gentler learning curve than proxmox or other hypervisors.