Clarification for binary search in solving optimal TSP when a polynomial algorithm with a budge exists
Input size refers to memory size needed to represent the input. In this case, it means number of edges + number of vertices, and you could also add size of the weights (which is logarithmic in their values).
No. It is written that we do $\log T$ calls and $T \leq |E| K$ (and this value can be reached), so the number of calls is at most $\log (|E| K) = \log |E| + \log K$.
Yes. It is never used in the algorithm that weights must be integers.
A $NP$-complete problem is a $NP$-hard problem which also needs to be in $NP$. So when speaking about a $NP$-complete problem, it is not false to state that it is $NP$-hard. However, $NP$ is a set of decision problems, therefore, TSP-OPT being an optimization problem cannot be in $NP$. We can still speak about $NP$-hardness as it can be informally defined for a non-decision problem as &quot;a problem more difficult than all problems in $NP$&quot;. When you say &quot;I thought that Traveling Salesperson Problem was NP-Complete&quot;, it refers to the decision version, denoted TSP in your reference.