Learn English – Push somebody over the edge

No, there’s no distinct difference between those. All can be used metaphorically or literally. On the edge of may be the most common for a literal meaning, and on the verge of for the metaphoric one, but one can find examples of all.

Less-common variants like threshold and cusp are also sometimes seen. There are also physical variants that don’t always work as well when used as metaphors, like margin, limit, or frontier, because these may carry other connotations, like being at the margins of society or the limit of one’s patience.

I think it amounts to meaning "too cerebral"—that is, too prone to considering multiple interpretations of data and the various pros and cons of possible reactions to them, without getting from that stage to the concrete response. Another way to express this criticism would be "too much thought, not enough action." It can also mean not giving due consideration to what other people are thinking or how they are likely to act—a problematic quality in a politician or strategist.

In contrast, "in over one's head" means "in a role or position that one is not competent (either by native ability or training) to perform satisfactorily."