Do I just replace hot/neutral with hot/hot?
Absolutely not! You have a vital step to complete before you can do that.
Note that the bottom picture would have been a little better if you had pulled the cord up so we could see the numbers on the white plastic block.
In this motor, you are working with two sets of wires, the electrical cord and a pair of jumper wires that go to the motor windings. These are the white and brown wire that go from the spade terminals on the white block down inside the motor (designated by arrows below).
Referring to the label, for conversion to running on 220/240V power:
The brown motor winding wire needs to be disconnected altogether. It gets removed from spade terminal #2 and pushed into what they refer to as the #6 pocket.
The white motor winding wire gets pulled off the spade connector on connection #4 and moved over to the spade terminal #2
The 240V power then needs to have each hot connected to it's respective LINE terminal in the pressure switch. The factory wiring between the pressure switch MOTOR terminals and the white motor terminal block for terminals #1 and #4 should be left alone
Basically, you are taking a pair of windings that run in parallel for lower resistance at 120V and changing them to run in series so the coil winding has higher resistance for 240V.
Note: Amended point 3, power input connects to the pressure switch, not directly to the motor...
To answer all the issues you raise would require a book on US electrical wiring. Or several. And a copy of the Code.
The vast majority of outlets in residences in the US are attached to branch circuits that are rated at 15 Amps and 120 Volts. Current practice and code calls for outlets like these
This version is tamperproof, required in many jurisdictions. The non-tamperproof look similar, but the slots do not have internal baffles
You may see different styles, such as Decora, or decorator style, which are functionally identical to basic outlets, but have a rectangular face
20 Amp circuits generally require slightly different outlets (if you are going to draw the full 20 Amps or there is only one outlet on the line) like these
But you can also find the lower 15 Amp outlets on circuits that are properly wired for 20 amps. Obviously 15 Amp outlets are limited in use to 15 Amp appliances, even if they are on a 20 Amp line.
Certain locations, especially where there is a risk of moisture, such as bathrooms, require a ground fault interruper (GFI) type outlet
These also come in tamper resistant and 20 Amp versions and vary like the basic outlets.
All of the above are grounded outlets, required in almost every jurisdiction for new construction and renovations. Some older installations may have ungrounded outlets.
These generally cannot be used except as a direct replacement for an existing one, and even then setting up a properly grounded outlet is preferred and may be required.
All of the 120 Volt outlets require a hot wire (usually black or red) and a neutral wire (always white). Grounded outlets also require a ground wire (green or bare). Outlets can be always live or switched. Live outlets have the hot wire coming directly from circuit without interruption. Switched outlets have the hot wire going through one or more switches before reaching the outlet so that the power can be turned on or off.
All of the 15-20 amp outlets shown above are duplex, that is there are two receptacles for plugs on each. These almost always are bonded together by a strip of metal. When you wire to one, both are energized. This bonding strip can be broken off allowing each of the receptacles on the outlet to be powered separately. This is most often done to allow one receptacle to be always live and one to be switched. This also allows each receptacle to be on a separate branch circuit (for heavy power use).
Some residences use higher amperage outlets for large appliances, such as an electric stove or dryer, and the outlets vary base upon a number of factors. Examples can be seen in the chart linked in the question.
Similarly, some residences use 240 Volts for large appliances and wells, and the outlets also vary considerably, and can be seen on the linked chart.
This is a very brief summary of the type of outlets most commonly seen in US homes. The full range of outlet types and uses is beyond a simple summary. The range of possible switching and wiring configurations also is nearly infinite. But this site welcomes questions on any particular configuration or problem you may encounter, so ask away.
Best Answer
In a 208Y/120 4 wire Wye system, you'll get 120 volts between either ungrounded (hot) conductor and the grounded (neutral) conductor. So if you're trying to connect a 120V appliance, all you need is one ungrounded (hot) conductor, one grounded (neutral) conductor, and one equipment grounding conductor.
The ungrounded (hot) conductor will connect to the brass colored screw terminal on devices (the small hole/blade), while the grounded (neutral) conductor will connect to the silver colored terminal (the larger hole/blade). The neutral conductor should never be connected to the equipment grounding conductor, or equipment grounding terminal on devices.
Notes:
Since it sounds like you're a bit over your head, you should probably contact a local licensed Electrician to finish the project and for any future electrical projects.
While you can solder connections in house wiring, it's not a common practice anymore. You'll also want to insure that you are using solder that is rated and labeled for the use, and that your solder joints are strong and insulated.