Do larger props = more thrust = more rotating mass (flywheel effect) = less power to maintain revs = less current draw?
All of the components of a power system contribute to your flight time. You usually won't get the best results by holding all the components constant and changing just one. So, for example, if you take a quad that has been optimized for 3S, and swap it for a 4S, the flight characteristics will change, but they probably won't be optimal any more. The motor and prop that work best with a 3S won't be the same motor and prop that work best with a 4S.
So what about prop size? In general, if you keep your motor the same and you increase the size of the prop, several things will happen. 1) You will get more thrust. Larger props produce more thrust than smaller ones. 2) The motor will draw more current, because it is having to work harder to spin the prop against the air resistance. You can see that there is a balancing act. As you increase the prop size, the motor's current draw goes up and thrust goes up.
At a certain point, you reach the motor's maximum current rating. What do you do then, if you still want to go to a larger prop? You switch to a lower kv motor. Because the motor is spinning slower, it draws less current, and it can swing the larger props. You could also switch to a motor with a higher current rating, but this approach is usually not preferred, because these motors are larger and heavier, and we usually want to keep weight to a minimum.
You asked:
Do larger props = more thrust = more mass = bigger drain = less run time?
Conversely, do smaller props = faster rpm = more thrust = less drain??
Now, you can see that the answer is not quite as simple as that. Larger props generally produce higher run times, but they have to be combined with motors that have an optimal kv rating, otherwise they will pull too much current and their efficiency won't be maximized.
And to improve run time, will a 4000maH battery = twice the duration of a 2200 maH?
No. Because when you add battery capacity, you add weight. Think of it this way: say you've got a 1 kg craft with a 2200 mAh battery on it. That battery has to lift 1 kg. Now let's say you add another 2200 mAh battery, which brings the all-up weight to 1.2 kg. Now you have 4400 mAh of capacity, but instead of lifting 1 kg, it has to lift 1.2 kg. So your capacity doubled, but your weight increased by 20% as well. If you were to double capacity while keeping all-up weight constant, then your flight time would roughly double, but usually when we add battery capacity, our all-up weight increases. So there is a law of diminishing returns in effect. You would see an inverse-exponential curve, with a plateau at the top where adding more battery capacity would add an infinitesimally small flight time.
Also bear in mind that thrust-to-weight ratio is going to come into play. If you have a 1 kg quad that is producing 3 kg of thrust, for a TWR of 3:1, and you add a ton of batteries until it weighs 2 kg, now the TWR is 3:2, and it probably won't fly very well. So you go back to the drawing board and pick different motors, different props, etc... this is why quad power system design can get complicated.
Will two 2200mah in parallel roughly equal same run time as one 4000mah?
Yes. The key parameter here is mAh per gram of battery weight. The two 2200 mAh will be slightly worse than one 4400 mAh, because there is extra wire and connectors, but the difference should be minimal.
Lastly, I guess its sorta obvious, but will a 3000mah 4s last longer than a 3000mah 3s??
In general, yes. By increasing voltage, you will increase motor RPM. This means that you will need less throttle to achieve a given amount of thrust. Assuming your flight behavior stays the same, you will fly longer on 4S than 3S. But to really make the most out of the system, you should do more than simply swap battery. Most of the time, if you switch from 3S to 4S, you will be able to change to a more efficient prop that will make the most of the higher voltage.
EDIT: And of course, that assumes that your motors can handle the 4S voltage/current. If they burn up because they can't handle it, then you won't get a longer flight time. Duh!
