what are the real problems with inline scripting? Is there a significant performance issue, or is it mostly just a matter of good style?

The advantages are not performance based, they are (as Michael pointed out in his comment) more to do with separation of the view and the controller. The HTML/CSS file should, ideally, contain only the presentation and separate files should be used for scripts. That makes it easier for you (and your peers) to read and maintain both the visual and functional aspects.

Can I justify immediate action on the inline scripting front to my superiors, when there are other things to work on that might have a more obvious impact on the site?

No, probably not. It can be very hard to convince the powers that be of pure maintenance work, even if it you believe it will save them money in the long run. In this case though, I don't think it is so important that you should stop everything and get rid of your inline scripting. Instead, just make sure that you make a conscious effort to rectify areas as you work on them for other reasons. Refactoring code should be something you do regularly but only a little bit at a time.

If you pulled up to a website, and took a peek at the source code, what factors would lead you to say "hmm, professional work here", and what would cause you to recoil from an obviously amateurish job?

The number one factor that would tell me it is not professional is the overuse of tables or divs. Here is an article explaining why neither should be overused.

what are the real problems with inline scripting? Is there a significant performance issue, or is it mostly just a matter of good style?

The advantages are not performance based, they are (as Michael pointed out in his comment) more to do with separation of the view and the controller. The HTML/CSS file should, ideally, contain only the presentation and separate files should be used for scripts. That makes it easier for you (and your peers) to read and maintain both the visual and functional aspects.

Can I justify immediate action on the inline scripting front to my superiors, when there are other things to work on that might have a more obvious impact on the site?

No, probably not. It can be very hard to convince the powers that be of pure maintenance work, even if it you believe it will save them money in the long run. In this case though, I don't think it is so important that you should stop everything and get rid of your inline scripting. Instead, just make sure that you make a conscious effort to rectify areas as you work on them for other reasons. Refactoring code should be something you do regularly but only a little bit at a time.

If you pulled up to a website, and took a peek at the source code, what factors would lead you to say "hmm, professional work here", and what would cause you to recoil from an obviously amateurish job?

The number one factor that would tell me it is not professional is the overuse of tables or divs. Here is an article explaining why neither should be overused.