When I extrude the face that's been highlighted in the first image, it overlaps with some part of the face marked as 2, as shown in the second image. How can I make it so when I extrude face 1, it'll also divides face 2 into 2 pieces so I can merge the overlapping faces. Or should I be following a different practice all together? Thank you for your time.
You should try to avoid overlapping geometry, like faces that occupy the same space creating z-fighting and other problems.
One of the many ways to go about this is to use loop cuts, and use those that define the places where the intersections should take place, and more importantly to create new vertices for such intersections.
Select one of the newly created edges, extrude to the point where the new vertex is (Enable snap to vertex for precise placement). Select the matching edge on the other wall and pres F to create a new face.
At the end of all of this do "remove doubles" to get rid of extra vertices