Club competition
Club competition adds another logistic to your race management challenges, so don't offer it unless you really think it will attract a lot of runners who would not have signed up without it.
Club divisions generally offer the following: Corporate (employees of the same company) male, female, and coed; and Club (non-employees, such as running clubs) male, female, and coed.
Coed teams should be defined, so they are all the same. Typically, they include even numbers of men and women.
Awards for club and corporate competition are given for first, second, and third place in each division.
To foster competition, don't let one entity (like one company or one running club) field multiple teams in each division.
For example, if a club fields fifteen men, they should all be on one team. The club should not be allowed to enter three teams of five people each. It should not be allowed to win 1st, 2nd, and 3rd place awards in the same division (although it's OK for a club to field a male, female, and coed team).
Instead, put all the men on one team and just count the scores of the top competitors (whatever number you specify to constitute a team).
If you are thinking of offering prize money for team competition, offer a lot, or don't offer any. A $100 prize, for example, is not likely to attract anybody who wouldn't run the race anyway. It would be better to spend that money on something that will benefit all runners.
The purpose of prize money is to attract elite athletes who will generate media attention. That will take a lot of money.
Also, don't charge clubs a team fee to compete. Running clubs usually can't afford it and corporations resent being soaked for extra money. Just charge each runner the regular entry fee.
Bottom line? Club competition is sometimes more trouble than it's worth. And corporate team competition is often not worth it at all.