Diablo 3 for the PS3 and PS4 will not feature cross-platform play but it will include an offline mode for PS3 and PS4 players. PC players cannot play the game without being connected to Battle.net - even when playing single-player. The clarification on cross-platform play was made by the game's community manager Vaeflare on the Battle.net forums.
"While we think cross-platform play would be awesome, there are currently no plans to allow connectivity between PlayStation Network and Battle.net (this is pretty standard for most games that have PC and console support)," said Blizzard's Community Manager. "As a result, the characters on your Battle.net account and PlayStation account will also remain separate."
On allowing Diablo 3 PC users to use a controller, Blizzard has no plans on doing that either:
"In terms of allowing an analog controller hookup for the PC, we don’t have any plans for that kind of support right now. Similarly, since Diablo III for console was designed with a controller in mind, the PlayStation version of Diablo III will not support USB mice or keyboards."
Both the PS3 and PS4 will allow four player co-op mode local multiplayer games. Players will be on the same screen instead of split onto four separate windows as well. This also means that single-player will likely not require a constant connection in order to play.