Specifically re 'forum' vs 'forums', that's largely down to the software's terminology as to what people use, I find.
SMF considers the entire installation to be 'the forum', and each area inside it is a 'board'. Other software, off-hand I know phpBB does the opposite, and IIRC XenForo too, such that 'the board' is the entire installation, and each area is a 'forum', meaning that 'forums' as a plural is accurate when you have multiple areas for discussion.
Putting it the domain can be good if you intend the site to be just a forum and never branch out too much. It can also be important if the word itself is part of the brand - like here. But if not, you're probably best to steer clear of putting 'forum' in the domain in case you later expand and don't want to have 'forum' in the domain because you have more than just a forum.