Given two path algebras $A$ and $B$, for example, A=: $1\to2$ B=: $1\to2\to3$, is the tensor product of A and B over a field $k$ a path algebra? if yes, how to represent it by a quiver? also, how to construct $(A,B)$bimodules?

It can be realized as a quotient of the Cartesian product of path algebras in question. You have to mod out by some commutativity relations. 


What you want is Basically you take the 1skeleton of the products of the quivers and you say that certain paths e.g. (edge,vertex)(vertex,edge)=(vertex,edge)(edge,vertex). 

