"Group" vs. "community" usage
Solution 1:
Well, when we say community, we refer to a large number of people or entities, who may be affiliated to many smaller or disparate sub-structures.Community almost always refers to a varied and large audience, yet with certain things, which maybe certain interests, opinions, or religion or ethnicity.
On the other hand, group is indeed most often used for a small number of people or other entities, each of which may or may not be large. Also, a group is not as natural a collection as community. A group maybe formed by some of us just now, but a community arises on its own and comes together, and has more naturally common attributes. Similarly, one can be expelled from a group, if he disrespects the rules, but one cannot be banned from a community per se, unless it is something very artificial like a virtual social networking aggregate that is moderated etc.
E.g. we talk about the international community of researchers of a certain field, say Physics, or the international community of institutional investors (each of which is an institution like a bank and consists of millions of people), whereas we say G8- a group of nations.
Solution 2:
community or interest group.
Not interest community.
Essentially, group is a non-specific collection while community has a commonality among its members. That is the reason group requires the adjective 'interest' and community does not.
You could think of a community as a group defined by locale/ common interest/ feature/ ancestry, etc. Btw, community is currently the buzzword.
Q2 Neither group nor community is necessarily associated with people as such. However, community is generally applicable to animate objects only, group to 'things' of all descriptions.
Q3 While group by itself does not have connotations of a small size, we have other words to refer to collections to suggest a large size, so that we do not usually see group used for large collections.
Solution 3:
One aspect is not just size, but strength of relation. When I hear community, I think either a natural occurring community of people who live near each other, or an intentional community of people who share many beliefs about how they want to live their lives, rather than a group having only a singular interest in common. Thus, the nuns who live down the road from me, though only numbering 8, is a community, and my town, which numbers in the thousands, is a community, whereas to me ELU is a group. Certainly, there are occasions when both could be used, and there some where neither would apply; I wouldn't call my church congregation either term.
Solution 4:
SIG - Special Interest Group is common.
Just to add to the befuddlement, there is also neighborship.