I was given (2) 400w metal halide pendants with the controller box. They each have a Hamilton double ended 14k bulb that are blown, however I have seen them both working in the past and know where to buy replacement bulbs. I have used a wide variety of LEDs and T5s but have limited experience with MH (I am aware of the heat implications, the cost of bulbs, and the need to change the bulbs every 12-18 months).
What are my options?
I'm not sure I understood your question here.
I believe what you are calling "controller box" is actually the ballast. Please post pictures so people can better help you.
What depth of tank would I be able to use this with and be successful at raising corals?
Down to about ~30" - 36" for most "photosynthetic" corals. But that, like any other light source will depend on what type of system, nutrients, placement, organisms, maintenance, etc...
Would one or both lights be too much for a 55g? 75g? 120g? 150g?
Depends on the measurements of the tank and the plan of the system.
We can't just generalize that. No matter what type of light system you get.
What type of corals would appreciate this much light?
Basically every "photosynthetic"coral would appreciate metal halide light. Again, the system you choose will need the right amount of light, so you need to choose what you want, then you plan the light around it.
If this is a viable option, should I purchase the 10k, 14k, or 20k bulbs? Thank you in advance for the helpful input!
Of course it's a viable option... for the right system!
It's better for you to make a plan and use the fixture for the system that you plan to have, not the other way around!
You can use the pendants with new ballasts for the wattage needed for that system you choose.
You can use the pendants and get 250W or 150W ballasts. Some electronic ballasts have a dial and you can choose the wattage.
10K, 14K and 20K ... will be more like your personal choice or the need for the corals you want to keep.