Except fishnet is used to analyze broadcasted games, puzzle discovery and others. Not used when analyzing your own games. You can easily test this by opening your task manager, monitoring your CPU usage, then opening a game you played on Lichess and toggling the aptly named "local evaluation". It's the toggle bar on the upper right, next to the text "Stockfish 14+ NNUE in local browser". You will see your CPU usage increase while the evaluation is on.

No, that is an analysis of a single position, not a game. The big button under every game which says "Request Computer Analysis" is done on fishnet as well, it literally says "Stockfish 14.1 server analysis".

That is true when you’re playing in your browser. On the mobile app (at least the iOS version) you can request computer analysis for your game done on the server.

You can request server analysis for your own games, up to 40 per day. They use fishnet.

<sarcasm> So ... time to implement fishcoins?</sarcasm>

Analyzing a chess game or two is not exactly easily verifiable, but it should be somehow possible to shoehorn that into a proof of work thing.