Are ad links automatically adjusted when a product URL changes in the product feed?

YES, the target URL is automatically adjusted in the ads.

However, this only applies to products. For all other elements (categories, brands and brand + category), the URL must be adjusted manually, as these are not sent via the product feed.

If your element gets a new URL, you should set up a "301 redirect" anyway, i.e. a permanent redirect. 

This is also an important SEO measure! If such a redirect is set up, the Google ranking is transferred to the new URL. If you have set up such a redirect, advertisements with the old URL would also point to the correct page and customers come to the correct page when they click on your advertisement.