Flipping all incoming/outgoing edges from a vertex in a DAG
This doesn't hold if we flip both outgoing and incoming edges as shown by @Yuval Filmus.

Here is my try of a proof by contradiction for only flipping outgoing of incoming edges (sorry if it's too informal):


After flipping all incoming(outgoing) edges from vertex v in a DAG G, we get another graph that is not a DAG (it has a cycle). 
Since we only changed the direction of the vertices incident to v, the only way we loose the acyclic property is if there is a new cycle going through vertex v.
Since we flipped directions of all incoming(outgoing) edges from vertex v, all edges incident to vertex v are now outgoing(incoming), so it is not possible for any cycle to go through vertex v, which contradicts 2.