Common drain stage/Source follower circuit analysis
I have personally never understood the appeal of 2 stage amplifier in the classical AB sense, and for that reason I never make them that way, but every single textbook has a 2 stage amplifier.  Regarding the output stage in the classic sense, I believe the reason is that you want enough output resistance for the feedback compensation to keep from ringing.  I use a cross coupled output stage that is similar to LMC6484.  You can get that output stage from the datasheet.

Regarding 1, the common drain stage is nice because it is biased to be at the edge of subthreshold so it have a very good linear behavior. The "charging" faster part is just due to the fact there's less total capacitance.  The band diagram looks like this (pulled from one of my lectures):


If you make the assumption that the devices are large so the threshold currents match, the result is: 
$$ I_{th}e^\left(\frac{\Phi_{sc1}}{U_t}\right)=I_{th}e^\left(\frac{\Phi_{sc2}}{U_t}\right) $$
This then can be modified with voltages based on the schematic above:
$$ I_{th}e^\left(\frac{\kappa V_{bias} -0}{U_t}\right)=I_{th}e^\left(\frac{\kappa V_{in} -V_{out}}{U_t}\right) $$
You can then push through the math and you get:
$$\Delta V_{out} = \kappa V_{in} -\kappa V_{bias}$$