How to test Eagle schematic design?
The only way you can really "test" circuits in Eagle is by using the ERC tool (Electronics Rules Check) which checks for shorts, misplaced connections, etc. You can find it at the bottom of the left-hand toolbar. What you really want is a simulator though, something like Multisim or LTSpice. I can't guarantee that either of those programs contain all the components you'll need though, so you'll have to mess around with it a bit. 

I am seeing quite a few issues with the schematic so far, but the ERC tool should point them out to you.