My photo was intended to show that it is impossible to tell a solid brass cover plate from a plated one (if they exist) in a photograph and beveled solid brass plates are not junk. The usual reason for wanting a truly flush outlet cover plate is if you think you will move your furniture and it will be in a walking path and not covered by a carpet. In my opinion that is a reasonable precaution and should be included in the design documents. The bottom line is: you should show what you want in the design documents and not blame an electrician for choosing an industry standard. A sub-contractor does not have the same obligation to anticipate or inquire about the owner's wishes as a designer would have. If the desired device and placement had been described in the design documents, the GC would have to correct the work at no cost to the owner. If only the placement is incorrect, you could probably get the outlet you want for the material cost when the location is corrected.

It sounds like the receptacle in the second photo is connected to the load side of the GFI (Ground Fault Interrupter) in the first photo, so that it has the same shock protection that the GFI itself has. A GFI is designed to open (interrupt) the circuit if it detects a ground fault (faulty device or cord "leaking" electricity that could electrocute a person). It can do that even if the GFI doesn't have a proper ground. If connecting the Smart Plug causes the GFI to trip, either the Smart Plug has a ground fault, or one of the Christmas light strings connected to the smart plug does. That means you have a shock a hazard. Someone could get hurt or killed. Unplug all the Christmas lights from the Smart Plug, and plug the Smart Plug in all by itself. If it trips the GFI that way, replace the Smart Plug. If the GFI doesn't trip with just the Smart Plug plugged in, then the problem is in one or more of the Christmas light strings. Carefully plug the light strings themselves into the GFI, one at a time. Discard any strings or other decorations that cause the GFI to trip. All that said, a lot here looks and sounds fishy to me. Maybe others have, so I welcome their notes on this, but I've never seen a furnace that was cord-and-plug connected. Every gas and oil furnace I've ever seen has been designed to be permanently wired. It would be a code violation (because of not following the installation instructions) to install such a furnace with a cord and plug. The only water heaters that can be safely and properly cord-and-plug connected are the ones that come with cords from the factory. These are always very small 120 volt point of use water heaters, a foot or two tall, with typically less than 5 gallon capacity. Also, that light switch is nothing like what I'd call a workmanlike installation. To me, that just shouts "hack work." Given such obvious poor workmanship, and your home inspector's report of ungrounded receptacles, I really think that for your family's safety you should hire a licensed professional electrician to look over your electrical system's safety.

I agree that a tripped GFI somewhere ahead of what you have is the most likely culprit. The challenge is to find it. Look behind furniture. Check in the cellar, especially right under or nearly under the bathroom. Look in the garage, including behind stuff. It might even be outside. When GFIs were new, they were expensive. Fairly often, a single one was daisy-chained to some or all of the other receptacles that the code said had to be protected. It's not common for a second GFI to be fed from a first, but I don't know of any harm that can cause. As for why it would be, it's possible that a previous owner didn't know the bathroom receptacle had GFI protection upstream (if it does), and added another. Another possibility is a loose connection on a receptacle that feeds the first dead one. That receptacle itself would still work if one of the downstream terminal screws were loose. Parenthetical anecdote: one of my rentals is a late 1950s tract house. I lived in it myself for a while, and during that time I did a little electrical work on it. I was astonished to find that on NONE of the original receptacles was the second set of terminal screws used. The electricans who wired the house -- and remember, this was not an expensive custom home -- actually made up soldered and taped connections in each box where there were receptacles downstream. The soldering was clean, too. No cold joints that I saw.

If you feel comfortable taking the machine apart, you can first unplug the machine and see if you can spin the motor shaft by hand (from what I can see on the parts diagram you will have to remove the pump to gain access to the shaft). The motor shaft should turn easily by hand. If it is seized, either something is wrong with the transmission or the motor's bearings are seized. If the motor turns easily, it is probably an electrical problem like the ones listed by dadoes. I would take a chance replacing the capacitor. Be careful while removing the old capacitor as it can hold a charge of up to 120 volts even with the machine unplugged. Make note of which wire goes where, and remove the old one and read the specs from it. Specs will read something like 350-385 mfd 120 vac. You can go higher on the voltage, but the important value is the mfd. It must be within 10-20% of the value of the old part.