Go to the PowerManage preferences and set Addon Sorting to As List.

This will make it a standard UIList, which can be searched, sorted, and scrolled through.

The addons list in the preferences uses a standard UI element. At the bottom of the list you should see a small arrow. Press it to open the sort and search options.

Yes, loading a preset will save preferences for the addons it disables, and load preferences for the addons it enables.

It does not backup or restore keymaps, because the API gives me no way to see which keymap items belong to which addon.

PowerBackup is a tool for saving and loading Blender and addon preferences in bulk.

PowerManage is a tool for managing which addons are enabled or disabled.

Either from the sidebar like all other addons, or from PowerSave's popover panel in the 3D view.

It just uses Python and the Blender Python API, so it should work everywhere.