Because combined with Burst Weapon that would pretty great with the Heavy Flak Grenadiers, but it seemed unlikely that a unit that has to Reload after every shot should be able to spend a whole turn on Sustained Fire, but I didn't see clarification one way or the other.

Sure it can use sustained fire, and yes, Heavy Flaks doing that against a unit that hasn't moved is OVERKILL.

Yes, just not on the same turn they reloaded.

And not indirect artillery fire