Will a node with no incoming arc or no out coming arc affect the maximum flow? So in a maximum flow question, say if we have a node which is not $s$ nor $t$, if it has no incoming arc, can we delete it without affecting the maximum flow? What if it has no output arc? How so?
Yes, you can. If it has, say, no outcoming edge, there can be no flow routed over this node. Otherwise the flow conservation constraint for this node ($v$)
$$\sum_{(u,v) \in E} f_{uv} - \underbrace{\sum_{(v,w) \in E} f_{vw}}_{= 0} = 0$$
is violated if you have incoming flow.