Online is fine with everything I've tried it with (Xbox, PlayStation, PC) and you can indeed play without an internet connection on all three. There are some little issues at times where the game will occasionally freeze, but nothing too substantial. It's not cross-platform, however.
Apologies for the late reply and thanks for yours. Great to hear this game can be played without an internet connection, often way too hard nowadays to tell which games need it and which don't. Think I will buy it for myself, now that I know it's actually buying and not renting. Thanks for the info, and also thanks for a fun game! :-)