My answers to your questions are in Red
1. How do I determine whether the main panel has the capacity for a 100 amp sub-panel? Note that I have two separate 200 amp main panels. It's a big house. A load calculation is needed to determine what capacity if any remains on the service.
See: Single Family Dwelling Electrical Load Calculator
2. How do I determine whether to use aluminum or copper wire? Price usually is the determining factor, due to large size and/or length.
What are the pros and cons besides price? Aluminum terminations need to be treated properly for reliable service.
Simply because of the length of this feeder, and the size service, you will be using aluminum, I am sure.
3. How do I determine what gauge wire to use? Need to size the feeder due to length.
4. How do I calculate voltage drop, and what is an acceptable drop rate? Need to arrive at the total demand load in amps, then you can calculate the size wire needed to reduce voltage drop. NEC recommends no more that 3 % for a feeder.
The formula is:
Voltage drop = 2 x length x amps x ohms per foot
You maybe able to an on line calculator, such as:
Southwire - Delivers Power
5. I'm planning to move the panel in about a year, but in the mean time, the panel will be only about 200 feet away. My plan is to use one 250 foot line and coil the additional 50 feet of wire next to the main panel until I'm ready to use it, and then just pull it through the conduit when I move the sub-panel further away. That way I won't have to splice the wire when I move the panel. Any safety or other issues with this? If I understand you correctly, the spare cable will be coiled up until you move a panel, and will be utilized until such time.
You should not use a feeder with 50 feet of it coiled. It will create heat, and possibly be a fire hazard.
You have asked some good questions, however, I am not sure you will completely understand what needs to be done. I highly recommend that you hire an electrician to help with designing this system.