Are there any metrics considering how heavily cited a paper is within another paper?
As far as I know, no.  The problem is that what you want to measure is impossible to accurately automate (even standard citation counts that Google Scholar automates are not 100% accurate) without hard-coding such information into the paper, as in Dirk's comment.  This of course has some issues, in addition to being a paradigm shift for the way citations are made (just as there are pressures now to include a citation, there would be pressures with such a system to make something an "important" citation).

The suggestion you made, about counting the number of times a paper is cited, crossed my mind in the past (along with some other ideas) but the problem is that it doesn't directly measure what you're asking.  There are different reasons to cite a paper, and the number of times you explicitly cite something in a given work depends on too many different factors, a large one being the author's writing style.

Remember the current paradigm: references are just that, references, which put the work in context for the reader and supply details that one uses.  Their purpose is not to provide a measure of the value of paper or researcher.  Once you go down the path of twisting things to measure worth and assign prestige (even if that is not your purpose), you just make things into a competitive game.