Are those normally run off of the prime mover's engine? I'd never even thought about that.

Every refrigerated container I've seen has its own cooling unit and does not use the trucks power.

I've always wondered about this. Do they have their own fuel tank and engine to run the compressor for refrigeration?

Yes.

And they have electrical hook up point for times when they are stationary. If you think it through, the time a refridgerated container spends actually connected to the truck is not all that big. It will spend more time in a port or on a ship, or even stationary at the departure point or destination.

All of the reefers I've seen have electrical hookups to power them when on ship/land and they have their own diesel powered generators to power them whilst attached to a truck.

There a a couple main reasons they have the diesel powered generator whilst attached to the truck:

1. Efficiency of the generator (truck companies don't want to spend more on fuel)

2. The truck is often stopped with the engine off (rest periods etc)

So in the short term attaching a refridgerated container to a Tesla semi wouldn't need and change in operation. In the longer term I guess it's possible that these containers will replace generators with battery pack to remove that emmission source, but they're likely to still be built into the container for the same reason outline above.

Could that eventual hvac trailer take regen power from the semi, given the extra weight.

It might be the case where battery in trailer makes sense.

They are using external diesel power generators, they are not running from the truck engine anyway.