I may have been misunderstood..? You were asking "what things PHP could do that normally JS would do," and I'm saying that, if you're looking at web development from a conventional standpoint, the question would be: "what could JS do that normally PHP would do?" Everything of a functional nature that can be done with JS, can be done with PHP (not vice-versa).

As for Flash, if you have a SWF that simply contains a button which goes to a URL on click, then there's no need for JS. And you can certainly still embed SWFs on a page without JS as well. But calling external JS functions from your SWF, would fail if JS is disabled, yes.

The ideal way to deal with your scenario - a set of link buttons in Flash that won't work without JS enabled - is to set up a div with your links in plain HTML, and then have JS replace the content of the div with your Flash. That way, if JS is disabled, the links that were originally in the div will remain visible, and if JS is enabled, they'll be replaced with your Flash content instead.

Also, as an aside: AJAX is not a language; it merely refers to a particular usage of Javascript.