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).
Rooms above garages are often done improperly. It's amazing, because it really isn't all that difficult, but a lot of people seem to mess it up.
Unfortunately, there's no "easy" way: You're going to have to remove some drywall to inspect and see how the insulation in the ceiling is done. There may or may not be a gap in the space, depending on how the ceiling was done. Even just taking the temperature above the insulation in that space will tell you a lot (above the insulation, it should be close to room temperature).
Drafts in that space are absolutely killer, it means there is a direct source of outside air. Black on the insulation is a good sign of airflow, indicating drafts.
Basically, if the ceiling space isn't totally sealed, the only thing you can do is turn the garage into a conditioned space (very expensive, in terms of both construction costs and ongoing energy costs), and even then, if the problem is between the insulation and sub-floor, it won't help.
What should be there at a minimum is a continuous vapour barrier, with a layer of insulation on top. The vapour barrier should be sealed to the subfloor or the vapour barrier from the walls above. The insulation should go all the way around the edge (in the headers), so the insulation is continuous from the floors up to the subfloor. There should absolutely be no drafts, exposed concrete, or anything directly connected to the outside that is uninsulated.
The best way (in my opinion) to seal this space is to use closed-cell spray foam, and form a continuous seal across the bottom of the entire subfloor. This gives you great insulation, no drafts, and acts as a vapour barrier as well (vapour barrier is not needed in this case -- though some building inspectors still don't understand this, so check your local codes). It also is better at ensuring fumes from vehicles in the garage can't get into your living space (in theory, vapour barrier prevents this as well, but 6mil vapour barrier is easier to puncture than a couple inches of hardened foam).
Though you may be able to find something obvious and fix it, be prepared that there's a possibility that the only "fix" is going to be to completely tear down the ceiling of the garage and re-do it properly.
If you're going to spend money and time on this, do it right.
Best Answer
What type of heating is used in that added on room? That will make a difference in how you solve the problem of the room being cooler than the rest of the house. Answer #1; yes insulate the floor if the basement under that room is unheated or cooler than the rest of the house. Answer #2; If the added room has 3 sides exposed to the weather, that is probably 1 or 2 more walls than any other single room in the house. If this is the case the room should have been built with more insulation than the rest of the house, since it will loose more heat due to the number of walls exposured. The type of heating system will determine how you correct the heating deficiency. You may have to add a separate heating device to solve the problem.