Open the dialer,press 3 dots on top right, settings and check the last option Always display ...

You might be confused about self hosting, your desktop should not be 24/7 online, you are the client and you want to use the vps as exit node, where you host also headscale. And please don't spend 20$ for a vps, you better rent 4 or more with 20$ from different providers and you can have more exists nodes. Headscale or wireguard are not eating a lot of resources. What you should look is at bandwidth , unlimited is better, you won't find many vps providers with unlimited bandwidth Even if you are behind a CGNAT using wireguard as client won't be any problems,because you are going OUT not coming IN.

At the moment im looking at AMD Ryzen 9 7900 and i5 13600k (if i decide to go with intel) i5 is 70-80euros cheaper in my country.

Is what he want to buy , not me.

Amd and Intel Just compare them, i will chose amd with closed eyes

https://github.com/fatedier/frp 10 minutes reading 5 to setup the best tool for tunneling i could find. The frps on a vpn(the most cheap you can find) and frpc on your server at home