Usually forum pages are loaded from several files. For example PHPBB has one file that is always displayed on top that is called "overall_header". It contains the login/ logout, register, profile, ect buttons. If you place the Adsense code in such a header file the ads will appear on all your pages.
Yes, but there is another problem that comes up with forums. For example, phpBB uses a Session ID to track a user. This causes the URLs to appear to be unique to AdSense and therefor end up showing PSAs. There are some mods that can be added that will help to reduce the number of pages with SIDs, but they can still be a problem.
You also need to be sure you are not placing AdSense on registration, error, and posting redirect pages, as it is against the terms. So placing the AdSense in the overall header of phpbb, for example, is against the terms (and yes, people have been warned for it).