1. Zeros in every cell.
2. If, and only if, the machine eventually halts, the number of 1s remaining on the tape is the score.
3. A contest to find the n-state busy beaver for each n.
4. An n-state busy beaver machine is one that is proven to have the biggest score of all n-state machines. A champion is the machine that currently has the highest score (a better one hasn't been found yet) but is not proven to be the n-state busy beaver.
5. If they didn't have the number of steps, there would be no good way to know if the machine ever stopped (the Halting problem is undecidable). If we try running it, and it doesn't stop, it might just be going for a really long time instead of forever. With a number, we know to stop trying after running the machine for the given number of steps.
6. No, there are multiple busy beavers for n>1 because of isomorphisms.
7. Because every step is required to have a shift.