What you need is pretty much a basic setup.
Your publicly available computers indeed should be setup in a DMZ. That is a third interface (originally called OPT1 unless you rename it) with a proper rule set.
If one of your switches is manageable you could use VLANs but the logical layout will be the same. Only physical layout would be different.
Avoid Realtek NICs if possible and go with Intels. If you need to troubleshoot something you know where not to look…
Depending on your friend's router is the choice of VPN. It probably does not support OpenVPN, otherwise give it a try. IPsec is not an alternative as long as both ends use dynamic IPs.
Since you have the hardware just go ahead and play around a bit. It's not that difficult.