Handle-ties have two reasons:
- Joint trip. This is relevant if a safety problem that causes one trip, would also require the other side be tripped. It's not applicable to multi-wire branch circuits (MWBC), for instance. In this case it is not applicable to dual 240V heater elements as that machine is designed.
- Common maintenance shut-off. Appropriate when we want to design the system so a worker de-energizing equipment is forced to turn it all off. This is required for MWBCs, for instance. It's not really relevant here, since it's readily obvious to any competent person that the heater is powered by 2-3 circuits, so he's going to make sure he shuts them all off.
In this case, neither instance applies, so handle ties between the 2-pole breakers are not warranted. Beware: the tripping half-breaker may not have the physical muscle to trip three additional half-breakers. So you don't do something like that without affirming the device is listed to do exactly that. Otherwise you violate the rule on using items only as listed (tested).
Using both grounds is a matter of why not, since it's already in the 8/2 cable. If it was connected via plastic conduit, that wouldn't apply.
In steel conduit, the conduit is the ground and no further grounds are required; you can run a ground of any size if you're really into belt-and-suspenders. I'm sorry, I made that sound weird. I would do that.