Should I cooldown before shutdown?
Depending on what material you print it it most likely a good idea to let the hotend cool down before shutting off the printer (fan).
For example if you shut down the printer right after you print a PLA part, at 190 - 220 degrees Celsius, your hot end will still be that hot and will suffer heat creep without the fan running. The next time you fire up your printer the hotend will be jammed and you will need to clear it before starting a print.
This is obviously situation dependent but in most cases you should let your hot end get below the TG (glass transition temperature) of the material before turning off the printer.
That option you are referring to, is meant to manually shut down power to the heated bed and hotend, there is no timed cool down period other than you timing it. This is a handy option if you fiddled with either the bed or the hotend; e.g. to insert new filament.
A cool down period can be very useful depending on the printer. Those cold-end cooling fans usually are very noisy, so people cut them of (that is not always possible/wise, but is being done out there), disable them after printing or shutting down power of the printer as a whole. Some type of hotends are prone to have heat creep up the hotend and soften the filament so that it can clog up the hotend. I've seen this happen on Ultimaker printers where the cooling fan was not spinning because some fine strings where sucked up.
To minimize the noise level of such cold end cooling fans you can put them on a relay switch and have e.g. OctoPrint schedule the print to be on for e.g. 2 minutes after a print failed or finished, works perfectly, and then you have your cooling down schedule/period.