Hi,
with voltage drop it's easy: the 3.7V supply is derived from 5V supply by dropping voltage across 2 diodes. Typical silicon diode such as those used in the circuit have drop of about 0.65V, so you get 3.7V. But if you look at the I-V characteristic in the datasheet of LL4007 you will see that at current of 2A the drop os almost 1.1V, so altogether you will get 2.2V of drop.
With mosfet the things are a bit tricky. The gate is typically at 5V and the source at 3.7 so Vgs is positive and mosfet is off. When the 5V rail falls below 3.7V the mosfet will start turning on it will connect 3.7V rail to battery effectively powering the 3.7V rail from it. Obviously, the mosfet will conduct from drain to source which is rather "non conventional" direction for a p-mos. It provides a low resistance path through the channel, but doesn't have any blocking capability due to parasitic body diode. The battery will also boost the 3.7V rail if it falls down more than about 0.65V below battery voltage (about 4.2 for a Li-Io cell). Clever, but a bit borderline design i must say.