The checks only occur at install time while downloading from their CDN (like freeShop does), and the checks that actually send console info that could get you banned occur while using the NIM module, and that module is used on sleep downloads.
In short, previously downloaded games are fine, and installing not-blocked titles (
here is a list)
not on sleep downloads is fine too.
The FBI thing you misheard is for when all (or most) titles are blocked by Nintendo and you can't download them through freeShop. When that happens you'll have to look up the .cia of the game and install it through FBI.
Playing online is fine, except for cheats and unreleased games.
I recommend updating, as you won't lose anything. The changes are made serverside, so staying on a lower firmware version doesn't help.