Ad code no longer in header.php file (after putting it there)

Can code you place in one file migrate to another one?

No.

I may have updated the theme since I originally placed the code. Can that do weird things in terms of making custom code in header.php disappear?

When a theme is updated, the entire theme folder is replaced — so any edits you made in any of the theme files will be lost completely. Using a child theme solves that problem.

But weirdly the ad code DOES still show up in the site’s source code, and the ads are still displaying.

Weird indeed. Can you provide the site address so I take a peek?