I have heard that Skyrim mods can leave behind loose scripts that clog up save files after uninstall if the mods are not implemented properly. It looks like I need to use a Quest script with "Start Game Enabled" to install my mod properly (currently for testing I use the console to add various features to my character, but obviously most people are not going to put up with that in practice). What's the best way to use a quest script to install a mod and properly clean up on uninstall?

There ain't one, kid.

The absolute best you can do is to make sure your objects aren't registered for updates and that you undo scripted changes to vanilla objects in the uninstallation.

You could try constructing your mod in such a way, that when it is uninstalled:
- Not a single vanilla script that may have been modified is running.
- Not a single new script introduced by the mod is running
- Any registrations that may have been made by the mod are unregistered.
- A save is made before uninstallation, after all the above are true.

That might perhaps mitigate some of the worst consequences of uninstalling

Still, there is no way to avoid all consequences, such as your save containing mod-related properties and variable data. No guarantees can be given, but if you are lucky, your game might then be able to at least partly recover from uninstallation, assuming, that your save might be free of stored copies of active mod scripts, that would otherwise continue to persist and wreak havoc even after uninstallation.