Does DFS in an unweighted DAG find the shortest path for each vertex from a source?
The only question is in the tile so I'm going to answer that.

A DFS visit from a source $s$ of a unweighted DAG $G$ does not find the shortest paths of $G$ from $s$.
As a counter example look at the following graph:


If vertex $a$ is visited before vertex $b$ you will not necessarily discover the shortest path from $s$ to $b$.

Also, the same example shows that the order in which vertices are considered by a preorder DFS traversal is not necessarily a topological sort.
Indeed, if $b$ is visited before $a$ the order would be: $s, b, a$ but there is an edge from $a$ to $b$.