Do careers only advance based on skill points?

Can they advance based on the same sorts of events that fulfill wants? For example, could a housewife career advance if you have a baby?

Career advancement is also based on number of friends. I can't answer the rest of your questions.

Career advances are coded into the career semi-globals. You can import those into your career clone and make them advance based on anything you can test on a sim (so you could test the number of times a sim has given birth by looking through their memory tokens, but not necessarily whether they have had a child in the last 48 hours). The downside of this is that the semiglobals for careers changed frequently between EPs, so you'd end up with an EP-specific career object.

If you look at the Harry Potter project, there's an example of a career which progresses on a non-standard property (in that case, the "Magic" skill token) which you could take a look at. It is EP specific though, so may not work that well in more recent EPs.

Thank you, that helps tremendously!