Questions about Farhi's pre-Adiabatic paper
Everything interesting happens in this paper happens within the 2D subspace generated by the two vectors $|s\rangle$ and $|w\rangle$. The vector $|r\rangle$ is a vector from this subspace orthogonal to $|w\rangle$, giving us the basis $|w\rangle, |r\rangle$, where the analysis will be very simple (using 2x2 matrices).

The norm in the second question is simply $|||a\rangle||^2 = \langle a | a \rangle $. The meaning of individual terms in the sum is: "Does the result of our (Hamiltonian) algorithm actually get away from the (useless) state we would get without applying the oracle at all?" If this term is small for many different instances (different $w$'s), the algorithm doesn't work, as it would likely give the same (wrong) answer for these instances. Thus, if the sum over all of the $w$'s were small, it would tell us that the algorithm fails. If it is large, it doesn't immediately mean it works, however, having the sum large is a necessary condition for a successful algorithm.