Hey there!

I do run a linux home server indeed. You can indeed use an old laptop, and many do, as they don't use a lot of power usually.

I am not sure about the nvidea shield, that would be cool. https://forum.syncthing.net/t/syncthing-on-nvidia-shield-tv-pro/20020 shows someone has done it. I'd be asking in those syncthing forums, people are very friendly there. That would be a great way to do it if it can handle it.

I hope that helps. If you want to get a handle on linux, run a virtualbox virtual machine first. I killed about 20 linux installs in virtualbox playing around before I went live on a real machine. It's a great way to find a distro you like also. I recommend pure Debian personally.

If you have further questions, I'll help as I can.