If I attack with a Celestial Apprentice and Friedrich Hemmler and my opponent defends, how many times can I trigger the Celestial Apprentice's ability? I'm thinking twice...

No. Once. Celestial says "other units in the same zone", so Friedrich deals 1 damage to each attackers and defenders, himself included. Friedrich is dealt 1 damage, so he deals 1 damage to an enemy unit in the corresponding zone.

If your opponent's defender(s) choose to assign 1 combat damage to Friedrich, then Friedrich would deal another damage in the corresponding zone.

Actually, Friedrich reads:

Forced : When this unit is opposed in combat, deal 1 uncancellable damage to all attacking and defending units.

This would cause one point of damage to both Friedrich AND the Celestial Apprentice.

Celestial Apprentice game text is:

Action : When a unit in this zone is dealt exactly 1 damage, deal 1 damage to target unit in any corresponding zone.

So, both the 1 one point of damage dealt to Friedrich and the1 point of damage dealt to the Celestial Apprentice should trigger his ability.

I this this is comparable to the High Elf location that deals damage each time a unit is healed.

Well, if that's what it says on CCelestial's text, so yes, it deals 2 damage when Friedrich's Forced effect is triggered.