I found this out the hard way too. First of all, I'm not sure if only premium elite clan members are counted towards clan challenges (for mw3), but this is a possibility. Second of all - the problem for me - challenges only include the people who were apart of your clan at the time you enrolled.
E.g. if you enrolled in all the clan challenges on the 1st of Jan, any members joining your clan AFTER Jan 1 will NOT be included in these. For this reason, you need to enroll day by day, if you are expecting new members to continue joining.
We think we can answer your questions (hopefully!), so here we go:
A couple things factor into competitions on Elite.
For Modern Warfare 3 Clan Challenges, non-premium members will count as long as there is at least ONE premium member in at the time of enlistment.
For Modern Warfare 3 Clan Operations, ONLY the top 6 premium members count.
For Black Ops 2, since there are no premium memberships, the top 6 members count for a Clan Operation.
No premium members need to be in the Clan's roster for Challenges, obviously.
For both of these games, only the members that are in the Clan before/at the time of enlistment will count towards the competition. This means that if you enlist in a competition, and then invite 5 people to your Clan, those 5 people will not count towards the competition.
Hope that helps! ^LM