[Edxu]

Our strategy sub-team is pretty small (4 members, 1 leaving this year), but we play a great role at competition in terms of communicating between teams and planning matches.
We've recently converted to a tablet scouting system, which has the benefit of giving us real-time data on the performance of a team at a competition, allowing us to plan matches better, using hard data as the base.

To answer OP's questions:

-Strategy Brainstorming
At the start of every build season, 610 watches the game release and then we break for approximately 2 hours while each team member reads the game rules. In that time, a collaboration between the Strategy and Administration heads create a rules quiz that each member must pass in order to participate in the build process.

People who pass are allowed to participate in the Textbook Strategy discussion, which is basically a round-table discussion between the entire team, led by the Strategy division, where ideas are thrown around and considered. By doing this as a team, we can get on the spot estimates of whether or not something is feasible, as well as getting the attitude of the team on a certain idea. For example, this build season, there were a few people who were initially opposed to our idea of being a container specialist without the ability to handle totes, but by calculating the maximum possible score with containers won on our side, the Strategy division showed that the middle containers were incredibly important to any Einstein-winning alliance.

The Strategy should drive the Mechanism, not the other way around. By designing a component that specifically fulfills a task, it's more likely to be successful, and it also gives your Mechanical guys a goal to work towards (eg this year: We need a 4-bin mechanism).

In terms of Strategy's purpose at competitions:
-Plan matches with Alliance Partners using scouting data off tablets
-Organize expert scouts and create a list of special traits to watch out for (This year again: opposing canburglars, strong stacking robots that synergize and robots that may have the potential to add canburglars for Playoffs).
-Explain match strategy to the Drive Team, making sure that they know where our Alliance Partners will be throughout the match and what they'll be doing. This makes it much easier for our Drive Team to focus on what they need to do without worrying about the rest of our Alliance.
-Walk around the pits and act as "superscouts" who look for tiny traits that may be useful in an alliance partner.

If you have any other questions, feel free to shoot me a message.