Ps4 – play on the PS4 while the girlfriend plays on PS Vita

Yes you can play PSP games online with the PS Vita so long as the servers are still up and running.

Also, apparently if the original PSP game supported Ad Hoc mode, Ad Hoc mode will work on the PS Vita as well.

https://support.us.playstation.com/app/answers/detail/a_id/3792

In order to pair up your Vita to your PS4, you will need to get both of them to pair up with each other. During the process, your PS4 will generate a code; you'll have a couple of minutes to provide your Vita with that code. Because of that, you will have to operate both devices concurrently; it's likely not possible without a partner.

Both devices also need to be on the same network, which should be achievable through VPN. Unfortunately, the PS Vita does not support VPN connections, although it seems that it can still use a VPN connection by sharing its connection with another computer (not sure if this actually works, so use at your own peril). However, I doubt Remote Play will be playable through a VPN connection, as lags and sudden disconnections are going to happen quite a lot; I've experienced some slight lags while playing right in front of my PS4 (which also happens to be right next to my wireless antenna).

One important fact to note, is that you have to use the same PSN account on your PS Vita as on your PS4, and whichever account your PS Vita has, will be the account it can be used on, on the PS4. In other words:
If your PS4 contains accounts A, B, and C (A being the PS4 Owner), your PS Vita must either have account A, B or C, and will only be usable for that very same account; so you can't play as A over your Vita, if your Vita has B.

As for the technology the PS Vita uses to pair with the PS4, it's WiFi. The PS4 even has an option to "connect directly to the PS Vita using a WiFi connection".