What version of AE are you using? If it's 12 or 13, I have some great news for you!
I am running version 184.108.40.206.
If you are limited by Element, I'd suggest using a much more powerful (though slower to render) tool that comes in CC versions of After Effects.
You have Cinema 4D included with the Creative Cloud!
And the extrusion is definitely able to be animated. Read more about it here: details of CINEMA 4D integration with After Effects CC and you can look through some tutorials there and here: Greyscalegorilla Blog | Getting Started with Cinema 4D Lite and Cineware
For what you're doing, you could put the texture on the face by choosing C1 or C2 as the selection tag for the material that has your image (this places it on the cap rather than the extrusion). If that doesn't make sense to you, you definitely need to go through some tutorials on using C4D.
That is good info and I will use it in the future but, I have a good workflow for the project now not needing C4D.