Placement of undervoltage monitor and cutoff
Q1. Option 1 - because the regulator output would need to be monitored for say 3.25V to allow some variation, and worst case temperature variation needs dealing with, and load transients may be an issue (or not) whereas monitoring at the regulator input achieves a "Minority Report" (Movie :-) ) effect of "detecting the problem before it causes a problem" and acting on a smoothly falling variable ( = Vin).

Note that 3.5V is well above safe Vmin for the LiIOn or LiPo battery and only the regulator dropout needs to be considered. 

Q2: Various issues. Needs more discussion.
Option 1 OK.
 Option 2 probably also OK. Has power down current considerations.