Explanation on some steps of AdaBoost.R2
Q1. $\operatorname{sup}$ is the supremum, or least upper bound.  You may think of it as the maximum, but relaxed in case the set of values doesn't actually have a maximum.  (And the pipes are just absolute values.)  https://en.wikipedia.org/wiki/Infimum_and_supremum
Q2. No, they mean $w_i \cdot \beta^{1-L_i}$.  See also https://stats.stackexchange.com/a/299266/232706.