Have you lived there long enough to compare the performance of the system to previous summers? (Sometimes units are undersized, or paired with poor duct design.)
Is the air coming out of the vents cold? Is it as cold as you remember before problems started? Is the airflow at the vents good?
After he added coolant, is the system working better? If yes, the system was low on charge. If not, something else is wrong.
It is possible for the system to be low on charge, but still be able to keep up on days that are moderate (albeit at a lower efficiency than a properly charged system).
It is also possible for there to be a leak, but the HVAC guy missed it. He probably only did a quick leak check (soap bubbles). There are more involved (expensive) tests he can do. However, they may not be necessary. If adding coolant made the system work well, wait and see. If the performance degrades again in a few weeks/months, you have a leak.
If there is a leak, it is usually in the outdoor unit, which is exposed to the elements, and takes a beating.
The good news is that the unit is probably still under parts warranty (typically 5 years for a cheap unit, 10 years for a good one), so if there is something expensive wrong, the parts should be covered.
(When was the last time your air filter was replaced? If it's been more than 6mo, look at it and see if it is OK. Replace if necessary. If you aren't up for replacing your air filter at least once a year, you should consider a service contract where the techs do it for you.)
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.
Best Answer
I would replace a 15 year evap coil. Since the efficiency of the condenser/compressor has changed (more efficient), you also need to match it with the proper evap coil and metering device (TXV).