If I can smell the trash in the garage from the room above it, should I be concerned about other vapors

Jonathan, your problem is very common. Do you know what is in the floor for insulation now? Assuming there is some f/G batts or something similar, it would be hard to try to add anything else, such as high density cellulose directly into the cavities. As much as you probably don't want to hear it, the best way is going to remove the sheetrock from the garage ceiling and remove the existing insulation. With the framing completely open, be sure to seal all holes through joists and floor areas where wires or other components may pass thru with foam. The absolute best insulation would be to have a high density foam sprayed into each bay and into any voids along garage door headers etc. This would give you the maximum R value and absolutely stop any cold air infiltration to the bedroom above. The goal would be to achieve an R value of 30 or more. The alternate method for DIY would be to again, seal any air passage ways and door headers with spray foam, then carefully install the max amount of f/g blanket and cover entire field with 4 mil poly and tape all seams and edges with tuck tape before reinstalling your fire rated sheetrock. This is not the best solution as the r value of the f/g is going to be limited by the size of the joists. 2X8 for example will limit you to about an R 23. A slightly higher R value could be achieved with blown in high density cellulose after sheetrock is installed, in place of the blanket f/g insulation. The spray foam is definitely the best overall, but the most expensive as well.

In response to Sharp Tooth's answer, too long for comment section: I have seen 2 to 4 inches of rigid foam insulation added to garage ceilings in this situation. The only drawback of this method is the problems it causes with garage door openers and suspension hardware. It also can cause issues with overhead lighting and overall head space. I hate to be a nay sayer, but I have to negatively comment on your other suggestions. Insulating a garage that does not have new style insulated, airtight doors is a total waste of money. Likewise, trying to heat a garage that is not build to be energy efficient can be extremely expensive to install and fuel, especially if a separate zone or plumbing needs to be added. This is not a viable solution as every time the garage door opens, out goes all that expensive heat. Carpeting, even over a thick foam pad will not help much either, maybe a gain of 2 to 4 "R", quite negligible. I hope you were not serious about heat tape under the ceiling. Other than electric or forced hot water radiant heat grids that would be installed under the floor, not under the ceiling, there is no safe product to do what you decribed. It would not work anyway, since the source of heat would be on the wrong side of the insulation and the vast majority of the heat would be wasted into the garage air. Sorry to shoot down your suggestions, but I truly believe questions should have informed, proven solutions, not wishful guesses.

I dont think there is much to worry about. Living in the snow belt in Ohio, our garage is almost completely black by the end of the winter. We try to brush or even spray it our a few times if the weather permits. But that is really to keep it from tracking into the house, not so much for the floor.

I would be more concerned with salt on your garage floor than the cinders. The salt can certainly eat away and break down the concrete.

There are concrete sealers that you could apply in warmer weather than might help to protect it, but I have never heard of anyone doing that in a garage.