You've already sorted this out, but the lesson here is simple: Don't ever assume anything.
Determining 220 vs 110 at the panel is fairly simple. You have a phase A and B in your house, which come from each side of the transformer. The transformer is center-tapped to ground, and this is your neutral wire.
"Voltage" is really just a measurement of potential difference between two points, and so if you think of it that way, the voltage between A and B is 220V, and the difference between A and neutral or B and neutral is 110V.
In your breaker panel, every other row of breakers is a different phase.
To get 110V power, you need a neutral wire, and a hot wire from either phase A or B (it doesn't matter which). So any single breaker, like #3, is only supplying 110V power.
To get 220V, you need one hot on phase A, and one on phase B, like the 220V breaker at the bottom right (#4/#5).
This picture also includes some tandem breakers (the one currently being installed is a tandem), in which case both circuits are on the same phase. That is, both #1 and #2 are on phase A. If you measure the voltage (potential difference) between #1 and #2, you'll get 0V.
Voltage between #1 and #4 is 0V, and between #3 and #5 is also 0V.
Voltage between #3 and #4 is 220V, since they're on different phases.
Note though, it's against code to use single circuit breakers to run a 220V circuit, since if the power trips, you want all the power to trip, not just one side (since that would leave 110V hot).
The other huge tip-off at the circuit is the wire colors (though again, never assume, always test). There aren't a ton of requirements for wire colors, other than two: white means neutral, green (or bare) means ground. If you use a white wire for something other than neutral (common in a light switch, for example), it must be indicated on both sides, usually by wrapping a piece of red or black tape around it. Black is hot.
The next most common color is red, which has two very typical meanings:
* switched, which you'll commonly see in duplex receptacles where one is constant, and one is controlled by a switch
* hot for 220V, which is the case here, where red is the opposite phase from black, to provide 220V.
To sum up:
- In the panel, a double-sized breaker on the circuit means 220V
- At the branch, not having a white neutral wire means you can't get 110V.
From what you have described, I see no reason why you would not have adequate space in your sub panel.
As far as wiring, you are correct in that you will have two hots (L1, L2), a ground, and no neutral.
Without seeing the spec sheet for exact numbers, I can only suggest that you take the conservative approach and wire using a double-pole 30A breaker with #10 wire. If you are using some sort of conduit then THHN will be fine, as you suggested.
Note that the NEC also defines rules for supporting conduit and whips. There is no way to answer this without knowing more about the physical setup and distances between your heat pump and sub panel.
Best Answer
You definitely need to hook up the bonding wire to that bonding lug. Right now it is hooked up to the earth ground, which is incorrect. Earth ground is to keep the pool equipment at earth potential, bonding is to keep all the pool equipment at the same potential. Run a bare #8 copper wire from the bonding lug on the pool heater to the bonding wire, and connect it using a split-bolt connector