The assumption is correct. Google doesn't care about a page extention. google_sitemap.php is fine.
About pages not listed..
It looks like your site is using a lot of hidden pages.
Meaning, you set the visibility as hidden, and use (manually created) links in the page to make them available.
The sitemap will not show hidden pages (you've set them hidden, so you don't want them to show)
The solution is to set them flagged visible. If you do not want them to be included in the menu, just use another menu for those pages (see
help on how to do that) and don't display that menu.
The sub-pages in your Faq are no subpages. It is one page with parameters. Following parameters is not something you should do in a sitemap.
This is also the reason that xml-sitemaps.com was not able to generate a sitemap. (it stopped at 500 pages in the sitemap) Links in the faq pages are even linking to the same page (sometimes with double parameters).
The faq.php page itself should be listed (as long as it is set visible) and Google will eventually go deeper in your site following sub-links in the page.
Ruud