The website requests an image or whatever from 27748626267848298474.example.com, where the number is unique for the visitor. To load the content the browser has to resolve the DNS for it, and the randomness ensures it won’t be cached anywhere as it’s just for you. So it queries its DNS server which queries your DNS provider which queries the website’s DNS server. From there the website’s DNS server can see where the request came from and the website can tell you where it came from and who it’s associated with if known.
Yes it absolutely can be used for fingerprinting. Everything can be used for fingerprinting, and we refuse to fix it because “but who thinks of the ad companies???”.
I think it counts. You always have the option of taking your data with you and go elsewhere which is one of the main points of self-hosting, being in control of your data. If they jack up the prices or whatever, you just pack up, you never have to pay or else.
Also hosting an email server at home would be an absolute nightmare, took me 10+ years to get that IP rep and I’m holding on to it as long as I can.
I have a mix of it: private services run at home, public ones run on a bare metal server I rent. I still get the full benefits of having my own NextCloud and all. Ultimately even at home, I’d still be renting an Internet connection, unless you have a local only server.