Well, first of all I should tell you, I checked around, and discussed this for you, and it looks like a Board group would have been possible here. I imagine a few wrinkles would have had to been ironed out.
For example if you were talking about a group who's function was to recruit members for an off-board site, I don't see that flying.
A board activities group organizing stuff like real time, streamed, events though, yeah I think we may have been able to help you with something like that. There would have been a number of possibilities.
But let's look at your other questions.
I have a question for the mods. Am I right in thinking that if we want Buttons and gifs relating to the SHPS in our sigs, do we have to register with you as a group? Cause if that is the case, the I take back my taking back of the request. We aren't doing this on another site, it's all being done through instant messaging, and we are planning weekly Buffy episodes. Would this be ok? Not what we are doing, cause that's seperate, but maintaining a small presence on the boards?
Let's consider the relevant rules in the FAQs here.
Please do not post links on the boards. This helps keep the boards orderly. You may include links in your signature, however. (But please do not include advertisements like "Come visit my site!" - it looks really tacky.) If you have a Buffy or Angel related site, you can add it to the links page. Please do not include links to other Buffy/Angel-related message boards as it is generally considered bad form to link to boards of the same subject.
So, if what you're talking about is a simple link to your facebook group in your signature, myself, I'd say that looks OK so far based on what that rule says, and what you appear to be suggesting. If I'm wrong though, I expect to be corrected here.
Moving on to the next relevant rule.
Board "families" are not allowed. Any creation of groups or clubs on this site MUST be cleared first with the staff.
This next one goes into something possibly connected to that.
Please also do not include lists of names of board friends - while we are happy that you have made friends on the boards, this can make other members feel excluded and is contrary to the community we wish to have here. Site staff may remove items from your signature at their discretion without prior notice.
Let's get specific about what's meant by a "Board family", or "clique". Specifically it's an exclusive group cut from the membership identifying themselves as special. Those are not allowed. Board sponsored groups are possible however. Examples of those would be Houses, and Discussion Groups. They're available to all and supervised by the board to make sure all can enjoy equally. Some exclusivity may be practiced by the board when members are recruited to contribute to a board service. An example of that might be the newsletter.
Now where does your off-board group fall into there. I'm not precisely sure at this point. I'd suggest contacting a mod via PM once you have the specifics worked out, and they'll bring the matter to the attention of the parties giving the final word on that one.