Can I swap out compositions, or control a composition's layer visibility via the timeline?
You can replace a comp in the timeline with a comp from your project tab:

Select a comp in the project tab
Select the layer in your timeline that you want to replace
Hold down alt
Drag and drop the new comp from the project window

It will replace the layer in the timeline, maintaining all the settings on that layer.
You can show or hide a layer using the Opacity control on each layer. Select the layer and press T to reveal them (remember it as &quot;opaci-T&quot;).
Then you can make a keyframe on that layer to animate it on or off (hold keyframes are useful for this).
You could also pre-comp all your layers onto a single layer, then use Time remap to select which frames appear when (and use the Loop expression on the time remapping to loop sections if necessary).