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If one has 2 batteries of 6 Volts each with 500 mAh each, then separately the batteries should provide 6 Volts * 500 mAh = 3 Wh each or 6 Wh total. However would it be the case that if someone where to connect the two batteries in a series that they would be able to double the Wh, as now the equation would be 12 Volts * 1000 mAh = 12 Wh. Why/Why not?

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The answer is no. The reason is the law of conservation of energy. The amount of energy provided by both batteries is always the same, regardless of how they are connected.

The way the batteries are connected determines the voltage and current at which they are discharging.

When connected in parallel both batteries are discharging at 6V, each at half the current required by the load (implying 6V * 1000mAh = 6Wh). When connected in series both batteries are discharging at 12V at full current required by the load (implying 12V * 500mAh = 6Wh). See also Kirchhoff's circuits laws.

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You got it wrong, the total available charge wouldn't change, so the result is 12 Volts * 500 mAh = 6 Wh.

Simplified a battery is spent when a certain amount of electric charge has passed through it, since you only get charge out of series connected batteries by the charge passing through both batteries the total available charge is the same as it is for one battery.

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