1. Yes
2. More like a club. However, our bot is not affiliated with my former high school since there are like a dozen high schools represented on the team.
3. Yes
4. Using Brandon's def the only debatable point is the physical activity. Barring the Human Player (that is obviously physical activity) I look at it with this quick, almost correctly formated (I dont quite know Modal Logic S5 rules) logical proof.
A. The robot physically exerts itself in the game. (motors pushing, lifting, etc)
B. The robot is created and controlled by my team. (autonomously and mentally)
C. Thus the robot is an extension of my team.
D. Therefore, My team physically exerts itself thru the bot
Also, FIRST is objectively scored, which is better than subjectively scored sports where a judge's opinion taints the competition. The competitors should be able to decide the outcome of a sport by themselves. When is the last time you played a pickup game and chose someone to be the judge to decide which team wins? Some subjective "sports" (I am not naming names) even award presentation points based on smiling, pretty uniforms, etc... I say "its not a sport if you have to smile while doing it, thats 'competitive art'" Anyway, if FIRST awarded presentation points, the Miracle Workerz would always win with our stylish green jerseys, hair and bots.

What kind of competition would that be?
5. We got a page about our yearbook about the assembly we hold every year. We also got a homecoming float after we won nats.
6. Use Brandon's def. minus presentation points.
Disclaimer: For anyone who plays presentation point "sports", realize this is just my opinion. I would consider those "sports" to be real sports with a minor judging tweak.