Like Bill said, check the installation manual and see what the minimum clearances are. If the minimum is 3" on the sides and you have 5" (which you don't) you could put 2" of foam (which I don't think you can).
I think that you might be good putting your foam around the front of the heater, where it comes though the wall. You may even have some extra clearance between the bottom of the cushion and top of the heater but I think the cushion is about as good a sound barrier as you're going to get.
In post #1, pic#2, you have a wall (at the top of the pic below the tape that says "5352"). You could put your insulation on the other side of that wall. The opposite wall looks like it has twice as the distance between the heater and the wall. If the minimum distance is the same on all sides (and TM adhered those minimum distances) you "should" be OK to glue foam to that wall.
Whatever you do, be sure that the adhesive you use is a good one. 3M has some spray trim adhesive that you pray on both surfaces, let it dry, then put them together. I know that stuff works.
If your foam happens to lose contact and fall against the heater, that could be a very bad thing.
Those vents are to keep the heat exchanger (inside the heater case) from over-heating the electrical circuit panels that are located back there. If they don't get enough natural air flow, the heater will shut-down from over heating (you hope).
The reason these heaters make so much noise is because they use a 4-blade aluminum fan blade on a small shaft. There is no noisier fan made than that. If you can find a heater with a squirrel cage type fan with a bearing on both sides of the fan (like most home forced-air furnaces have), that would be much quieter.
Please keep us posted on what you find out. I may like to do the same thing. The thing about sound deadening is that you don't necessarily have to "block" the sound. Sometimes sound deadening put in the right place can "absorb" a lot of sound.
BTW.....the pic in post #4.......I may be wrong but I think the guy glued the sound deadening foam on upside down. I'm pretty sure that the open cell part of the foam is designed to absorb the sound. I think that the smooth side is designed to be glued against a flat surface. It seems to me that the sound would just bounce off of that shiny smooth surface. That would pretty much defeat the purpose of using sound deadener. I would think that the sound entering the open cells of the foam would get trapped in the foam and die (thus the term sound "deadener")