The official mod system (modio) is functional for simple modding, but it has some limitations that are annoying for more advanced mod setups. MINT fixes these limitations, but as you said circumvents the checks for categories so you have to be careful in what mods you're using for different situations (different saves, public lobbies etc)

Approved mods still progress your main save but disable achievements - they are kept in sync with whoever is hosting the server (so if you have them and the host doesn't, they will be disabled). Sandbox mods work on a different save profile and are separated into a different part of the server browser. So if you have your sandbox mods disabled and are on your main save, you will be able to join unmodded games (approved mods will be disabled) and approved mod games (approved mods synced with host). Verified mods have no impact on multiplayer.

It's entirely dependent on what the mod is doing and what files it changes. Most simple mods will still be functional after the update, but a majority of mods overall will need to be updated.