Concrete – Heat causing concrete to heave

You have a couple things going on here: 1) The posts. an engineer would have to evaluate to be sure, but it's most likely that you will not be able to remove the posts without providing beams of some sort to span the space between the walls. Their irregular arrangement probably lines up with walls that they are supporting in the first story, which in turn are supporting the floor of the second story. 2) Clearance. Digging deeper to get additional clearance will be challenging, because you will have to leave the footers under the existing foundation in place, and not undermine them. This means you will likely end up with a floor that is a fair amount smaller than the existing space. Also, you will have to support the part of the house that are held up by the posts while they are being relocated/lengthened.

In either case, you will need advice from someone knowledgeable in building structure and foundations.

Overall, it may be easier/cheaper to raise the level of the house and add additional height to the existing foundation (if it is in good enough shape to take the additional load) than to dig deeper. A contractor that is experienced in this will know how to support the house while it is being raised, and you may be able to include new support beams to eliminate some or all of the posts at the same time.

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.