When I begin to fill the pan with the batter do I fill it to the top?  I've been reading some have had issues with their bears missing a portion of their legs/feet.

When cooling, can I leave the core in so that nothing will collapse in?

The cake batter should come to top after you put the core in.  Make sure your pan is well greased in all the nook and crannies, even on the rim. Also the inside and outside of the core, and rim. I let mine cool on its back. Use a firmer cake.