If you have ANY traditional AC system this is a terrible idea, it is allowing a significant amount of conditioned air to naturally transition outside.
If you already reliant on natural air flows and fans, can time it right to feed cool air in, and use some insulated panels to manually block either/both sides not in use it will help. Also, fans are typically (manufacturer specs can greatly differ) more efficient the larger they are, the longer the blades the more curved they get because they are laterally spinning faster at the same RPM; so if you double the width you could place a fan on each half.
Even with this, seal and insulate all areas possible. Most caulking is relative easy to remove so I'd caulk the fan in and remove during the cold season. Insulating and blocking off windows that you don't mind blocking, like bathroom windows already with heavy privacy shades or ones facing the non-sunny side. There are plenty of additional, and better, ways but I can't think of any other right now like these that you can do to a rental and on a budget.
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
Where to find the specifications
Listed on the furnace
Somewhere inside the service panel of the furnace, there should be a schematic for the furnace. This may list the electrical specifications of the blower motor. If not, the blower motor itself will have a label on it. It should list the voltage, amperage, horsepower, etc.
Using an Ammeter
An ammeter is used to measure the amount of current flowing through an electrical circuit. You can use a clamp on ammeter to measure the amount of current being used by the blower motor while it's running.
Simply set the meter to measure current (A in the pictured meter), and clamp it on the hot wire feeding the motor.
This method may require working on or near live wires, please use caution and/or consult an expert to take the readings
Calculate Usage
With the above information and Ohm's law, it's easy to determine how much power the motor will use over a given time period.
Ohm's law says that
Power (P)=Voltage (V)xCurrent (I), so we can determine how much power the motor uses at a single instant in time.ex.
Next we have to determine how many kilowatt hours the motor will use, which we do by dividing by 1000.
So for every hour the motor runs, it will use 0.943 kilowatts of power.
Finally, if you multiply this number by the amount you pay per kilowatt hour. You'll see just about how much it costs to run the motor constantly.
Notes:
Results using this process will be an estimate only.
Values used are not actual values, they are example values only.
Calculations may be incorrect, nobody checked my math.