If the outdoor fan comes on and the compressor does not, you need a "hard start kit" this is either a potential relay and start capacitor as in my diagram or a PTC thermistor solid state start assist device that piggy backs onto the run capacitor. This is assuming your compressor is cold to the touch and has not been starting. If your compressor has been starting and is off due to the thermal overload being open Then it is due to the capacitor being bad or refrigerant being low or the compressor is mechanically tight or electrically shorted to itself internally, this is not something a homeowner will usually be able to do unless walked through it step by step.

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.