I'm wondering if I can print objects that are made up of separate pieces. For example, printing links of a chain together. If so, I assume I would have to create the original model with the pieces 'floating' so they wouldn't be touching each other.

You are correct. Although some materials can actually be printed with extremely close tolerances, and placed very close together, it is better to model your design with spacing between the parts such as rope chains or chain-mail or even bracelet.

Moving parts can be achieved with very little effort. Just make sure you leave spaces between the parts.

Some materials cannot be made with moving parts easily. The best materials for moving parts are metals, and any SLS or Selective Laser Sintering process (powder-bed method).

More pictures will be posted with help on this topic soon.

I suggest making a group in the community to discuss this topic specifically too.