You don't have to worry about a heat pump being inadequate for cooling tasks, they are available in a large range of sizes. For really large buildings, they can be ganged together. 800 sf is a walk in the park for residential sized units.

I don't know the specifics of what determines when the backup furnace kicks in. IIRC, it is the combination of a lower thermostat set point and the outside temperature. It kicks on when the pump is not keeping up or it's too cold for it to run efficiently. I can assure you you will not be sitting for hours waiting for backup heat. No customer would sit still for such a system.

While it is true that heat pump heat is free, you are just paying to move it around, if you've had heat wave cooling bills before, you know moving heat around is far from cheap. Only a careful analysis of the various options, considering both operational and installation costs, as well as life cycle replacement, can you make a proper decision about which system is most cost effective. The correct solution will vary by small changes in energy costs and climate.

In cold climates, a ground heat source system should at least be considered because it will reduce or eliminate the need for backup heat. If you are concerned about your carbon footprint, do not forget most electricity generated in the US is by burning fossil fuels. Your footprint from using a heat pump will vary greatly depending on where your local power is coming from. Heat pumps make a lot of sense for many people, especially if they are investing in cooling equipment anyway. But everyone's situation is unique, so the best solution can be arrived at only after careful rational analysis of all the factors.

Disconnect the power

Start by turning off the breaker, and pulling the serviceman disconnect, which will typically look something like this.

This will insure no electricity is flowing to the condenser unit while you're working.

Open the unit

Next you'll want to disassemble the unit, to allow access to the electrical parts. This will vary from unit to unit, so check the owners manual for the procedure for your unit. Once you have the unit opened up, make sure to discharge the capacitors.

These things store enough power to kill you, so you don't want them to discharge accidentally.

Resistance is not futile

Once the power is completely removed from the unit, it's safe to start poking around (electrically speaking, don't go busting the refrigerant lines). Start by tracing the wires from the condenser fan motor, back to where they connect in the electrical box. There should be 3 or 4 wires. In my unit, I had Black, White, Brown, and Brown with a White stripe (your model may vary). To determine if the motor is good, you'll measure the resistance across each coil. To do this, you'll have to disconnect the wires, so the motor is no longer part of the circuit (make note of where the wires connected).

Typically you'll have 3 wires, start, run, and common (we'll ignore my 4th wire in this answer). Set your multimeter to measure Ohms, and start measuring. You're going to measure the resistance between each combination of two wires to determine what each wire is, and if the motor is still good. Let's start with Black and White...

Black -> White = 15.9
Black -> Brown = 35.4
Brown -> White = 51.2

Knowing that...

Common -> Run = Lowest resistance
Common -> Start = Medium resistance
Start -> Run = Highest resistance

We can determine that...

Black = Common
White = Run
Brown = Start

If we also know that the two lower readings should always add up to the larger reading, we can safely say this motor is still good. If you measure 0 or infinity between any pair, that means you have a shorted or an open winding and the motor should be replaced.

Repeat the same procedure for the compressor motor.

Shorts on the ground

The other thing you'll want to check for, is shorts to ground. Set your multimeter up to test impedance. Put one probe on the equipment grounding conductor of the feeder, and the use the other to find a solid ground on the motor. You may have to scratch some of the paint off, especially on the compressor. Once you've found a solid ground, measure from each motor wire to your ground spot. If the meter beeps or give a low resistance reading, you have a short to ground. As with the resistance test above, the motor should be isolated from the circuit when doing this test (once a solid ground is located).