Trim Paths on shape layer converted from Illustrator

It looks like you want to just apply Trim Paths to the outlines of the face. If that is the case your Illustrator document should have all of the lines you want to reveal on a separate layer and there should be no brushes applied to any paths and no fills. Just a simple stroke. 

When you import as a comp you should only convert the layer with the paths you want to animate to a shape layer. When you convert a vector path to a shape layer you can end up with a lot of extra junk in the layer that is not needed. If you start by pressing 'uu' and looking for a way to simplify things. If all you want is to animate a stroke and the stroke is the same size and color for every path you can eliminate everything but the paths and the stroke.

 I see this kind of question fairly often so I threw together a very quick tutorial on the process. 

There is nothing preventing you from adding a separate new stroke for one or more of the paths, adding individual Trim Paths or making groups, but most of the time, just eliminating everything but the paths, one stroke and adding a shape layer animator like Trim Paths is all that is necessary.

In order to see the Trim Paths in action you must add a Stroke thickness, I only see the path but I don't see Stroke, give it more thickness and have a test modifying the start and end parameters of the Trim Paths. If you don't want so many groups, try to have a single complete path from Illustrator, not several pieces of paths.