A FiveM server by itself can only use a maximum of 4 threads IIRC, so you’d be looking for a setup that has fast core speeds (3.0+ GHz generally) to run the actual game server; make sure to get ample RAM, however do note it doesn’t have to be particularly fast - DDR4 will suffice - and it doesn’t have to be strictly server-spec. Lastly, make sure you’re rocking enough broadband width, a good target to aim for is 2-3 MB/s of true speed per player.
I’m not familiar with hosting TeamSpeak servers but I cant imagine it takes a whole lot of resources to run.
Either would work okay as long as the specs are adequate, there’s no big difference TBH. A dedicated server is always preferred, of course, as you have sole access to it, but there’s nothing super wrong about getting a VPS if you want to cut some costs.
I’m pretty sure that’s not the case, one of my FiveM servers right now is using 115 threads, another one is using 267 threads. Overall, 3 FiveM servers have spawned 12 processes (4 each).