You’ve got a pretty good grasp of the concept of nesting compositions. Here’s how I would specifically go about this project:
Create a precomp* for each of the individual body parts, each of these compositions only needs to be as long as that body part’s animation. Then put each of these pre-comps into an intermediary comp (another pre-comp) and loop them with time remap and a loopOut() expression. You should then have all three body parts looping through their respective animations in this intermediary comp. Make this intermediary comp 60 frames long (60 is evenly divisible by the number of frames in all of your looping animations). Then place this comp into your final output composition and loop it.
FYI, the answer to ‘Should I build these with precomps vs comps?’ is: both. A precomp is just a name for a normal composition that is nested in another composition, the precomp functionality in AE allows you to automatically nest layers into a new composition that will then be placed in the composition you pre-comped from (confused?). For example, if you imported a circle from Photoshop and placed it in your main comp, then decided you wanted it pre-comped you could use the pre-comp function in AE. If you already knew you wanted it ‘pre’-comped then you could’ve created a new composition for the PS circle, placed it in this ‘pre-comp’ and then put the ‘pre-comp’ in your main comp.