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.
It sounds like you may be mistaken as to how this is wired, or that perhaps I'm just not understanding your explanation. As others have mentioned, it's not possible to get 240 volts from a single pole in a 120/240V split phase system. Each tandem breaker provides 2 120 V circuits, this is true. However, if you measure between the terminals on a single tandem breaker, you'll get 0 volts. This is because the terminals are both powered from the same leg, and so are at the same voltage potential. If you measure from a terminal on the top tandem breaker to a terminal on the bottom one, then you'll measure 240 volts. This is because each breaker is connected to a different leg, which are each one half of a 240 volt circuit.
With all that said. For this setup to work, one appliance would have to be connected to both breaker. Something like this...
Notice that each appliance circuit has one wire connected to each of the tandem breakers. In this situation, you'd need a device like Speedy Petey shows.
Which ties the breaker handles together, to provide common trip characteristics.
Notice how the inner handles are tied together, and that the outer handles are also tied to each other. This way if either trip (or are turned off by the user), the entire circuit is shut off.
If this is wired the way you've explained, where the dryer is connected to the top tandem and the heater is connected to the bottom. Then there's some magic going on in those breakers.
Best Answer
Probably not, but check the nameplates to be sure, and a neutral can be a good idea anyhow
For a dedicated circuit, the power requirements are determined not by any paragraph in Code (it's legit to have a 240V-only dryer or range/oven circuit if said appliances don't need 120V, or not have circuits for a dryer or range if they run on gas), but by the nameplate on the appliance the circuit feeds. So, you need to check the specs on what you're putting in to be certain you're running the correct circuits for them.
However, for the common-cases of the appliances you listed, you're likely to be correct -- electric dryers and ranges generally require 120V for lamps, controls, and the drum drive on a dryer while running the heater from 240V, while the rest of the appliances in question run their controls either directly from 240V or from a Class 2 control transformer, and all the motors in them are 240V as well. Granted, running a neutral alongside the 240V may be a good idea, even if the appliance doesn't use it, so that 120V receptacles and luminaires can be provisioned for service purposes -- this is particularly important for the well pump and compressor/outdoor unit, but may be wise for the hot water heater as well.
(P.S. the trick with the neutral is to replace the normal disconnect box with something like an 8 or 12 slot subpanel -- this lets you have circuits for lights, service receptacles, and the like to go with your well pump, compressor, or whathaveyou without having to pull a zillion wires out that way.)