Basic questions about network flow calculations
I think all your questions are answered by examining the actual algorithm to compute the max flow.

With regards to your Q1: in the max flow problem your input is a graph and capacities on edges, your output is the max flow. Thus, max flow is a function of the graph and capacities.

The basic algorithm for computing max flow is as follows. Your capacities are $c: E \rightarrow \mathbb{R}$. We shall incrementally compute the flow $f: E \rightarrow \mathbb{R}$. Let $f^i$ be the flow in iteration $i$. To begin with set $f^0 = 0$. In iteration $i$ construct the residual capacities $c^i(e) = c(e) - f^i(e)$ - this is how much capacity is left on edge $e$ given that we are already sending $f^i(e)$ flow through it. Find an augmenting path from $s$ to $t$, i.e., any path $p$ from $s$ to $t$ such that all edges on that path have positive residual capacity. How much flow can you push on this path? Clearly it's the minimum capacity along an edge of that path $f_{add} = \min_{e \in p} c^i(e)$. Update your flow by pushing $f_{add}$ flow along path $p$. This process naturally stops when your residual graph does not have an augmenting path from $s$ to $t$. It's a fact that at that point the flow you constructed is, indeed, the maximum flow. This fact requires a proof. This is a meta-algorithm (called Ford-Fulkerson), because I have not specified how to choose augmenting paths. Certain choices are better than others. Using BFS (in which case, this max flow algorithm has a name of Edmonds-Karp) to choose the augmenting path results in a polynomial time algorithm, but it again requires proof.