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.
This is a complex problem, and it's going to be difficult to provide an accurate answer without inspecting the equipment. So this is going to be more like a few comments, than an actual answer.
Water heater
I'll start with the water heater.
Old units aren't efficient
If the unit is older, it's likely not performing at peek efficiency, which is wasting energy. Updating the unit could save you money in the long run.
Tank heaters waste energy
No matter how much hot water you use, the heater is turning on from time to time to heat the water in the tank. The only way to eliminate this waste, is to install an on-demand unit.
Fuel source cost
Depending on the costs of different fuels in your area, switching fuel source could save money. Installing a gas fired on-demand water heater, could save you money long term.
Fireplace
If the fireplace doesn't have a fan, that is blowing heat into the home. You're quite possibly drawing more cold air into the home, than the amount of air the fireplace is heating. By using the fireplace, you could be blowing money out the chimney.
Also, the cost of wood is typically quite high, compared to other fuel sources.
Heat pump
Heat pumps use a lot of electricity, which in many parts of the country is an expensive fuel source. In temperate climates, heat pumps make sense because they're not typically used much. In climates where the temperatures often drop below freezing, heat pumps are virtually unheard of. This is because heat pumps aren't so good at heating homes, and often have to rely on supplementary electric heating elements.
Use the furnace on cold days
Since you have a gas furnace in the home, you can use that when the heat pump can't keep up. Depending on the local gas price, this could save you some money. In most dual fuel systems (like yours), if the heat pump can't keep up it shuts down and the furnace runs instead.
From your description, it sounds like the furnace and heat pump are running at the same time. This is not a good idea, and can actually cause damage to the system.
You'll want to configure the system (or have it configured) so that when the thermostat switches to auxiliary heat, the heat pump shuts down and the furnace kicks in.
Without actually being at your home, it's difficult to diagnose the problem. You might want to have an HVAC technician come out and evaluate the system, to help determine if you're wasting money.
Best Answer
In theory, yes, you could repurpose an A/C unit to be a heat pump but that's not a good choice. In order to switch between A/C mode and heating mode, you would have to re-plumb the lines every time you switched between heating and cooling or fashion some sort of reversing valve mechanism of your own design to accommodate this. The American Standard 4A7A6048J1000AA is NOT going to have a heating mode since this installation also has a gas furnace and the extra cost to put in a heat pump where it's not needed is just wasted money. Retrofitting is going to be a complete custom job and you're pretty much on your own doing something like this.
Not only that but it's going to be extraordinarily inefficient since the unit is designed as an A/C not as a reversible heat pump system.
The best choice, in my opinion, is to use your installation as designed, the A/C unit for cooling and the gas powered furnace for heating. Gas heating, especially in TX, is going to be significantly less expensive for heating and electric powered heating.
If you are set on eliminating the need for gas, the best approach is to replace the A/C unit with a heat pump system that would efficiently perform both heating and cooling.