Why are number of landings logged? I have flight logs for a WW II B-24 navigator who flew with a combat crew in China Burma India.  There is a column for number of landings.  The entries range from 1 to 5 or 6, mostly 1 or 2.  The time can be as long as 8:00 to 10:00 or more.  Why is the number of landings recorded?  Is that still done today?
Yes, it's still done today, at least in military aviation.

It's to ensure that the undercarriage does not excede the number of landings it has to endure before being checked/overhauled/replaced. Landing an aircraft can typically be the most stressful regime of flight in terms of loading of certain bits of the aircraft, so it is pertinent to log how many times it happens per flight.