The board should have reverse polarity protection for both battery and external source...
My Question: The FET (FDN340P) and diode (1N4001) work as reverse
polarity protection for board?
In this circuit the MAX6326 is powered directly from the external source. Absolute maximum input voltage rating is -0.3V to +6V, so reverse polarity will kill it. And you require the circuit to accept 12V, which will also kill it. There is no simple solution to this problem.
Reverse battery protection is also iffy. If the battery is connected in reverse with external power off then the FET's internal body diode will stop the reverse voltage from getting to the output. However if external power is then applied the FET will turn on for at least 100ms until the MAX6326 comes out of reset. During this time a very large current will flow from the external power supply into the battery. The components will probably survive this momentary surge current, but the high stress could reduce reliability.
Given the problems with this circuit you will have to look for other solutions that don't involve a MAX6326. The simplest is to just use a Schottky diode in place of Q1. If you choose a high current Schottky the voltage drop could be less than 0.3V. If this is too much then just add another cell to the battery to make 4.5V, then a standard silicon diode will do the job.