I'd be hesitant to run it at your own house. While you can use a cloudflare tunnel, I'd never expose anything in my home network to the outside.
Digital ocean is cheap, there's another called hetzner which looks also pretty cheap. So you start will rent 1 core VPS for 5 bucks, it's enough to run your own instance but not really enough to host any communities.
If your server is local, meaning it is on your network, then you connect via the local netowork. If you are both wired, then you'll likely be connected via 1Gbe. If you server is wired, and you connect wirelessly, you are limited by the wireless connection speed. If your jellyfin server is remote, meaning physically hosted offsite, then you will be limited by your internet speed and the speed of the jellyfin servers connection.
If the connection is local and its dropping, check out your jellyfin servers resources to see bottlenecks. Also see if you can check your tplink router and see if the CPU is spiking.