Current path question
The current from the 10 V source won't follow the yellow arrow because that doesn't lead back to the power source. Current only will flow from a voltage or current source through a circuit if it can go back to that source, so that a closed loop is formed.



This schematic may help you to understand this. It's exactly the same schematic as in your question. So why would the current coming from R3 go to the left part of the circuit? It's a dead end, there's no return path.

Kirchhoff
Let's call Kirchhoff in. Kirchhoff's Current Law (KCL) says that the total current in a node must be zero; the sum of the incoming currents = the sum of the outgoing currents. 



Applied to node A this gives us I1 + I3 = I2. But when we follow I1 through V2 and R3 we see that it's the same as I2, since there are no branches on the way. So if I1 = I2 then I3 must be 0. Even more simple when you know that KCL doesn't only apply to nodes, but to any closed boundary:  



The sum of all currents going out of the blue rectangle = the sum of all currents coming in. There's only one possible path, so I3 must be zero to satisfy the criterion.