And some answers...
1. No. 1953 Kaisers used a woven mat that had the appearance of being a grass mat.
2. The toggle switch controls the under dash lighting. Most switches quit working long ago.
3. Questions 3 and 12 are related. The box under the dash is the power supply and tuner for the radio. The piece in the dash is known as a control head. It's just the dial and buttons with cables and wires that lead to the larger box. This is an unusual set up and not common to the 53 cars. There's another frequent user of this site who goes by 'Fid'...he's our radio guru and I'm sure will go into detail about this particular unit.
4. The antenna can be mounted on either side, though in 1953 the passenger side seems more common.
5. The filter looks like a Purolator unit, although the strap appears to be for a larger filter given the interesting adapter. Purolator units were used but the Fram canister was much more common in the full size Kaisers. Purolator units were used in the Henry J's as well.
6. The carburetor looks like the right unit but the Mileage Minder is an aftermarket unit. There should be a brass tag near the carb throat that will have either 781S or 999S embossed. The 781S was used in 1953, although either will work on the car.
7. Your car has factory installed electric wipers, picture 11. In 1953, the Manhattans used electric wipers so the fuel pump is incorrect. The broken pump is probably the right one. The pump shown is for cars with vacuum wipers.
8. The cap looks like it has 4lb printed on it. If so, it is the right cap.
9. No, you are not missing anything. If your car had a standard transmission with overdrive, the relay would mount in that spot. No, overdrive means the holes stay empty.
10. That's the turn signal relay. Turn on the signals and listen and you'll hear it click.
13. That is a fender skirt and yes, they are held in place using the spring loaded arms. There should be two small square clips mounted above the wheel that the top of the skirt slides in to, as well as two inverted U shaped tabs that accept the spring loaded arms.
Hope this helps!