Further, what if my war has a dependency on the core? Would I include it under WEB-INF/lib?
For regular libraries (i.e. jars not containing any managed components like EJBs etc), you will have to place them in .ear/lib directory if those libraries are to be accessible to all components and (other libraries) within the entire .ear. So in your case, you will have to create a lib directory under the .ear and move the commons-pool.jar into that directory.
You should retain the core.jar at the top level of the .ear since it contains deployable components like EJBs.
That really helped! The server appears to be discovering the dependencies and loading the different deployables correctly.