Everything you listed are applicable to BuddyPress. I’ll briefly go over your points below:
(1) allow the imbedding of media in “Group>Forum” posts as can be done in regular posts?
You have a few options here. You can type in manual HTML as Scott stated or you can look into finding a bbPress plugin that handles uploads like GD bbPress Attachments.
(2) allow the posting of a post within Groups as opposed to restricting the posting to just “updates”. This would also allow the inclusion of media?
(3) allow individuals to post regular posts under their profile, as opposed to simply updates? (same reason)
Activity updates are meant to be simplistic like Facebook and Twitter. However, you are able to embed links to popular media sites like YouTube by simply pasting the URL on its own line.
For more info, check out this article:
It applies to BuddyPress as well.
However, if you want to embed media by uploading, you can look into some BuddyPress plugins (such as BP Media, BP Gallery, BP Activity Plus), however please note that we will not be offering 3rd-party support for them as we have not vetted these plugins for inclusion into CBOX.