In general should the 20/40 coexistance box be checked?

In general, yes, it should be checked.  Wi-Fi standards require that routers support 20/40 coexistence, which is functionality to ensure that routers fall back to using 20 MHz wide channels either when devices request it or when neighboring Wi-Fi networks that overlap with the current channel are detected.  Disabling coexistence can interfere with these functions.

Note, that the 20/40 MHz coexistence mechanism only applies to the 2.4 GHz band.  Unless you live in a rural area with no other Wi-Fi networks present, there's no point trying to use 40 MHz channels at 2.4 GHz.  A 40 MHz channel uses 2/3rds of the frequency spectrum available at 2.4 GHz, which is very unneighborly.  You would be better off using the 5 GHz band, whenever possible, because 40 MHz channels are allowed, even 80 MHz.  You also can take advantage of 802.11ac Wi-Fi which uses superior modulation schemes for overall faster speeds.