Secondly, if a 1:1 transformer is used between the meter and the load, would the meter still read correctly?

If you put the transformer between your electric meter and your referigerator, of course the meter is still going to measure the power consumption...

AC just looks like a sine wave on an oscilloscope so what characteristics of that sine wave are measured to determine wattage used?

If you have a resistive load, like a light bulb, the current and voltage are in-phase. So, you can simply measure the RMS current and the RMS voltage, and multiply.

With inductive or capacitive loads, the current and voltage can be out-of-phase by some amount (up to 90 degrees). In that case you have to measure the voltage and current at the same instant and calculate the power for that instant, or take many measurements or measure continuously to get an RMS power measurement. (I'm sure you use some calculus in your physics class, and you might remember how to multiply two sine waves with a known phase-angle difference...)

If you call an electrician and ask him to measure the power your refigerator is consuming, he might not have a power meter handy*, so he''d probably break the circuit and insert his multi-meter in series to measure RMS current. Then, he'd multiply by the known line-voltage to calculate the power. That wouldn't be 100% accurate, since the compressor motor is somewhat inductive, but it would be a reasonable approximation.

If you wanted to do someting similar with the Arduino, it would be easier to measure the peak current. Since you know it's a sine wave, you can multiply by 0.707 to get the RMS current. And, since you also know the (approximate) line voltage, you can calculate the power.

If it's for a physics class or for the power company, it's very important to measure the true-accurate RMS power consumption. For casual/hobby use you may be able to take some shortcuts.

*Power meters are somewhat rare, but I assume most electricians now have a

Kill-A-Watt in their truck.