I used pkhex on 3ds since US/UM, And on switch since SW/SH till now SC/VI and so far i havent been banned. Even if some pokes cant be placed on home, so far safe
I use jksv (same as checkpoint) to backup/restore saves and send to home. As long you use a legal game there shouldnt be any issue, even if you get the dlc pokemons without the dlc purchased
I have also some homebrew installed, and going online without any issue, I don't have any illegal game, everything was legally purchased and haven't got any issue so far
i dont know about achievements but i believe there is a sepcial tab for that.
and yes, you can send your pc island to mod switch and then to normal switch