I recently did not do so well in a couple undergraduate core classes, what are my options to make up for this on a grad school application?
I will answer the first question here.
If the course you're skipping is a "mandatory" course in your major, then taking advanced courses to make up for the earlier failures is not sufficient unless you get the necessary approvals to do so. (Usually there is some sort of departmental advisor whom you can petition.) However, I should mention that given your past record, it is quite likely that such a request will be denied.