@ObsidianUrbex @gme
I very recently just setup Iceshrimp.net, but the main reason I chose it is unlikely to be important to your use. I wanted to be able to react to posts with custom emojis :3
gotosocial seems like a solid choice, but you need to bring your own front end that is compatible with the mastodon api. I set it up before I realised this was not suitable for me because it lacks custom reactions. If you only use the app that shouldn't be a problem, and rrees.me/2024/03/24/alternative-mastodon-frontends/ has a few web front ends.
I imagine most servers have a setting for how long remote media should be cached, for example gotosocial the default is 7 days. This won't affect the photos you upload.
docs.gotosocial.org/en/latest/configuration/media/#__codelineno-0-105
As for federation I think that servers discover you when you follow someone on that server, or someone on that server follows you. I don't think a relay would help people find you but it could help you find others. But I don't really know for certain.