Easiest/Cheapest way to improve insulation of a room
For Q#1, no. It is typical, indeed expected, that they will only insulate the parts of a wall that border the outside of the home.  Insulation isn't that cheap.

For Q#2, get an IR/no contact thermometer, frequently selling for $20 online for the lower end models.  Check all the walls in the room and compare to other walls on the same side of the house.  This will give you an idea of whether the problem is real or imagined.

The reason I suggest the issue could be imagined is that people, lights, and electronic equipment put off a lot of heat, possibly more than you are seeing transmitted through the walls.  Just by working in the room, you will likely see a large spike in temperature.

The easy fixes for this problem are more weather stripping around leaking windows, outlets, etc and adding insulation to places you can reach, such as the attic.  And finally, make sure your air vents are properly adjusted to send more air to the rooms you use.  You may consider placing a fan on the floor at the door blowing cool air into the room (effectively pushing hot air out the top of the doorway).