Thanks for the detailed response, Henry!
As far as the approximation of battery life, I found a site where other people have actually figured out a pretty accurate formula for it. With that, and another site that calculates Ohm's Law, I've done some math(yes, my head hurts), and found the battery numbers I've been looking for. From what I can see, without any other type of draw on the battery, a 3.7 volt, 750mAh battery would last about 0.255 hours. Doesn't seem like much, but they also give you a way to translate that into actual use time:
B*3600/A=N
Battery life(B) in hours multiplied times number of seconds in an hour(3600) divided by average activation duration(A) equals number of activations before battery dies(N).
So on the given battery above, with an average activation of 5 seconds(my previous estimation was WAY off), that is 183.6 puffs.
Now, a new question: If I use a 9v battery, and use PWM to create an output voltage of 3.7(ish), would that improve/increase the battery life? My brain is telling me that if I'm turning the power on and off like that instead of running continuous voltage, I should increase the life by equal to the percentage of battery voltage I'm NOT using(i.e. if I were using 4.5 volts from a 9 volt battery, it should last twice as long as a 4.5 volt battery under the same load).
But I'm also as I said, no good at math. Can you confirm or disprove this theory?
On another note, bought my Arduino today. Decided to go with the classic Uno. Didn't have cash on hand for the Mega, but if all goes well, I'll be adding one of those to my collection eventually.