A popular multiplayer cheat is to suspend the network stack somehow for a split second while their team gets into place and resume causing the game's network code to resync and players will teleport into advantageous positions or become extremely difficult to target or hit during key moments. It's very hard to detect especially if the program doing the suspending is on the router and not the system running the actual game.

Example of this in action: https://youtu.be/I5QyNA6H6rY

Sure, the issue is that even if you don't use it, the simple fact that it is running in the tray may be enough for an anti-cheat to trigger. Just something to keep in mind.

That doesn't just apply to multiplayer, though. OP said Diablo 3, which is multi OR single player. Many games now use those phone-home anti-cheating measures (or drm measures) even when you aren't actually playing with anyone else.

Diablo 3 is an always-online game; the reason is that you can take the items you earn in single player and use them for your multiplayer character (and vice-versa). Their security is extra tight because up until recently, said items could be traded for real money. And you don't want anyone to be able to generate things like that offline.

Although they did, there were some item duplication bugs. And even that was irrelevant, since there were quickly popping up small armies of item / gold farmers; every once in a while they'd get banned, but by then they would've earned enough money to buy another fifty copies of the game and continue.

Is Diablo III single player? When I played it at the release, connecting to a remote server was required (which is a shame, btw. I will miss the modding community that Diablo II had).

It can be single-player, but you still need a connection to do so.

Is this also true for the console port(s)?

It is multiplayer only (always connected) but you can play it alone.