I'd want the light switch to be at the "standard switch height" for your house. That way there is no searching the wall for the switch in the darkness. Or out of habit. I'd not want the towel hanging on the ring/bar to cover anything up. Nor would I want the wire from whatever is plugged into the outlet to be in the way of any switch or to get tangled with the towel. Will things be permanently plugged into it? Or will it be "time of use" only? That can make a difference. With the Thermostat most likely being the least used and if it can't be set at the same height as the light switch, I'd set that a bit higher up on the wall than the light switch/fan switch combo, but aligned vertically with it. Personally, I prefer fan/light combo timer switches. A single switch controls both. You turn the switch on, the light and the fan come on. You turn the switch off, the light turns off immediately but the fan runs for an additional preset time. If you did that, you could put your fan/light switch and outlet in a dual gang box at the appropriate switch height. Put the thermostat in its own single-gang box a little higher on the wall, but aligned vertically with the switch box. Below that, at a height high enough to keep your draped hand towels off the countertop (if I understand your situation I think it's hand towels), but low enough to deconflict with the outlet, place the towel bar/ring. I'd use painters tape and mock up the locations and see how they work. Unless I'm really misunderstanding your description, I don't think "how they look" will be an issue. "New things" sometimes look weird to the un-used to it eyeball. Once you get used to the look, all will be well. So prioritize substance or functionality over style.

On the WhiteRodgers website, that model is obsolete. The replacement (-0211) uses batteries, or at least has batteries as an option. Have you had a thunderstorm lately? Could have been a nearby lightning strike fried it. When you turn 'it' off, which 'it' do you mean? The thermostat or the power to the AC at the breaker panel?

Check with some of the Civil War associations in your area - they can be a goldmine of information. The Library of VA has a special section for Maps & Photos & I suspect other States do as well. If I remember correctly, the original Union maps are housed at the Nat'l Archives in MD(?) and the State Library had copies of local areas but these were not allowed to be photocopied. Apparently, the Union made detailed maps (even the outbuildings) of the enemy & my house happened to be surrounded by the White House of the Confederacy, the South's largest Army Hospital, the port leading to the Atlantic, & a major munitions foundry. I saw a portion of the map in a book & started from there, only to find out I needed to go to the National Archives to order a very specific section of a specific page of a specific map & I've never spent the time nor the money to do it. Because Richmond was the Capital of the Confederacy, perhaps the residential maps were more detailed but I suspect a lot of areas were covered - maybe not so many north of the Mason-Dixon. Try googling Civil War maps, National Archives, & your area & see if anything pops.

The problem is upstream. Check the other receps on the circuit for a bad connection. Plug a table lamp that you have turned on into the non working recep. Then, while someone watches the lamp, go around an plug something into each upstream recep and wiggle it around. When the table lamp flickers, you have found your bad connection. It will most likely be the neutral(white). Another possibility is a GFCI that you think is reset but it is not. The ones with the black and red buttons do not always reset easily, sometimes not at all. Since this is an outdoor recep, depending on when your house was built, it should be on a gfci, maybe the one in the bathroom? And a third possibility, are you sure the recep is not switched? Your test results indicate an open neutral, could be caused by any of the above (including a switched neutral).