The DNS registrar Hexbear is using now is called: Porkbun
As for server hosting:
I use Digital Ocean for their most basic tier server, which costs me about $6 a month. However, I only use this server as a VPS (Virtual Privet Server) Gateway and Reverse Proxy. The server hosts a WireGuard VPN that connects back to a box I control that is stuck behind a CG-NAT. The VPS Gateway is necessary to pierce the CG-Nat and allow traffic to the box on the network I control.
On that box is where I host all the things I want to host. If you're not behind a CG-Nat then your network should have a public facing IP address that you can tie to a domain... However, I'd still recommend using some kind of VPS Tunnel, to not expose your ISP provided IP address to the world.
If your network gets a new IP address, you'll want some kind of DynDNS (Dynamic DNS) type service that can automatically update your DNS record with your new IP address. Though, again, if you're using a VPS Tunnel, then that shouldn't matter either, since your local machine will connect to your VPS as a VPN client, while your VPS acts as the VPN host.