Can I do this with Jellyfin?

[TheDreadPirate]

Downmixing: How well does Jellyfin handle downmixing 5.1 surround sound to 2.1 stereo? The audio boost setting on Emby doesn't work that well and I constantly have to double/triple the volume on my Klipsch speakers.

Depends.  If the audio is transcoding, there are server side settings for how that happens.  Including choosing a downmix algorithm and whether to, and how much, boost is applied.

Dashboard > Playback > Transcoding > Audio boost when downmixing and Stereo downmix algorithm

Client side downmixing is pretty basic for most clients.  The desktop client, Jellyfin Media Player, uses libMPV as the playback engine and accepts MPV configurations.  Meaning you can set custom MPV configs, including downmixing and boosting audio.

Users in another thread came up with the MPV settings you would add to Jellyfin Media Player.

Settings > Client Settings > scroll down to the bottom.  There will be a big text box to paste MPV options.

https://forum.jellyfin.org/t-stereo-down...1#pid47741

For the other clients, I am not aware of a way to boost audio within Jellyfin if the client is locally downmixing.

EDIT: Forgot to mention that Jellyfin MPV Shim, another desktop client, will also accept the same MPV downmix/audioboost config since it is pretty much a wrapper for MPV.

Recursive Discovery: I have an elaborate folder structure to organize my TV shows that I refuse to deviate from. All of the file paths for my TV episodes end in Series Name\Season#\Series Name - S00E00 - Title. But, every series is in a Genre folder which means I have to manually add each genre folder to my TV Library in Emby. Then I have to do the same thing for my Anime Library. If I rename a Genre folder or add a new one, then I have to manually add them again. It's demoralizing. I want to add 1 folder to a library then Jellyfin discovers every TV show regardless of how many nested folders I use.

Jellyfin behaves the same as Emby in this regard.   Can't imagine why that would be.  

Tree Node Sidebar: I want to see each of my libraries in the sidebar and I want to see a carrot dropdown that reveals the library's genres and their respective media.

Search our feature request site for what you're describing.  If you don't see it, submit a feature request.

https://features.jellyfin.org/

Custom Top Bar: When you select a Library in Emby, there's a horizontal menu that has buttons named Shows, Movies, Genres, and Folders. I would like something similar, but I can control the options or maybe replace it with a genre menu.

The top bar is already a thing, but not customizable.

   

Multi-Star Rating: The one thing I miss from Plex is the 5 Star Rating feature. In Emby, you can only Favorite something which is objectively dumb.

Jellyfin is the same as Emby in this regard.  There is an existing feature request.

https://features.jellyfin.org/posts/1134...al-ratings

Write Tags to Files: I know this one is a long-shot, but it would be great if I could write ratings and basic tags to music files. No expectations here.
Thanks for any help!

AFAIK, there is no functionality in Jellyfin that will modify your media files, period.