I do not think they stop forward movement but that is purely anecdotal from attacking and being attacked and pursuing/chasing where run speeds 'seemed to match'.

This isn't really a question about in-game physics, but yeah I'm pretty sure that Regulus's auto-attack at least can be executed without stopping.

The only one that can't fight while moving is Fire TB.

OH yeah, you're right over there, frost TB can move and shoot at once, whilst in frost mode, but not in fire mode, almost forgot about that.

Yes, he has to stop to play with his pole.

You will keep autoattacking without slowing down, as long as you stay in range (need same or better speed to do that). It looks like there are a couple of exceptions to this (see posts further up), but this is the general rule.

If you use any ability with a cast time, you will stop while you cast, and go into idle mode when you're done casting.

If you use an instant ability, your demigod will use it and go into idle mode. Once in idle mode, the "idle AI" will decide to attack the closest target. Since you're still close, it might attack the enemy demigod, but I don't think it will chase (can't recall right now). You will have to be fast and r-click him again to be able to pursue. You can also experience a known bug, which will keep you from being able to attack-click the demigod again after having used an ability until you have a issued a move order first.