Electrical – Understanding electric heaters power

Right. I've seen this before. Some handyman put in a subpanel, but only ran 120V service to it (who knows why - saving money on wires? Expanding one wire feeding an outlet into a whole sub-panel?) and then jumpers the hots - which works OK if everything is 120V loads - which was probably the case when the panel was installed.

This does, of course, mean that a 2-pole breaker has pretty much 0 volts across it, as both sides are fed from the same phase (both are 120V to neutral/&ground.)

You'll need to add another hot wire from the source to the panel (traditionally red in the US system - black hot, red hot, white neutral (grounded), green or bare safety ground.)

Depending what the current wiring is, and especially if it's inappropriately sized (which would go right along with the handyman's decision to run it 120V) it may be more appropriate to replace the current wiring completely with 4 wire cable (called 3-wire cable due to the convention of not counting the ground wire) or 4 wires in conduit.

I'd also double-check the status of the ground and neutral - in a sub-panel the neutral is supposed to be isolated from the ground (including not being connected to the case, which should be grounded) and I would not bet on that being how this one is, just following the pattern of behavior shown by whoever wired it up.

You should probably go over the whole electrical system looking for any other "interesting" work that's been done to it, and possibly call in an electrician to check you on anything you can't sort out. Incidentally (clarifying my comment), both QO and Homeline are fine, it's just that the breakers have to be the same flavor as the panel, and sometimes folks get the wrong one as both are Square-D products, and they almost-sort-of fit. But that has nothing to do with your problem, from your description in the comments.

1. No. You may risk damaging your unit by running it in cooling mode when it is too cold out (although some systems have safeties around this), but heating in low temperatures should not damage the system.
2. No. A correctly functioning heat pump should not ever drop below 100% efficiency.
3. This is dependent upon the specific heat pump model. Often, manufacturers provide charts/graphs describing the efficiency at different temperatures, but I do not see one available for yours. Two data points are specified; at 47f it is rated at a COP of 3.5 (or 350% electrical efficiency), and 17F a COP of 2.42.
4. No, but the heating capacity is often close, in mild weather. Yours is rated at 33,400 BTU/h at 47f and 21,400 BTU/h at 17f. Measuring this requires two temperature sensors (before and after the coil) and an accurate airflow measurement.
5. Your system defrosts on a timer, after 30/60/90 minutes of compressor run time, depending on a jumper setting. (this is less efficient than smarter systems that detect increased air pressure from from frost blocking air movement through the coils).
6. Protecting the unit from falling ice from eaves, and from accumulation of snow, is a good idea. However, make sure to leave ample space so that the airflow isn't restricted.
7. It says on the side that your blower draws up to 2.3 amps.