There are no absolute answers but 
Q1. Do the batteries charge when I am moving down the road?
A1: usually yes, however this is not 100%, only way to be sure it to monitor them for a trip.
Q2. When I am camping without electricity and using the batteries how
long will they stay charged?
A: Depends on how much battery and how much electricity you use its all math If you put a steady 5 amp load on a 100 amp hour battery than about 10 hours. 
Put another battery in parallel with it, the hours MORE than double.  Increase the load and .. Well that "More than" applies both ways.
Thus the bottom line is: We can not answer that.
With limited use of lights and furnace they **SHOULD** last at least 8-10 hours minimum.
Q3. If I run the generator to charge the batteries when I am dry
camping how long should it take to fully charge the batteries?
This depends on your batteries and converter.. Ideally the converter should be about 30% of your battery bank's 20 hour rating.. IE for a single Group 29 (About 100 amp hours) 30 amps.  This will take about 2 hours to bring it up to 90 percent and 2-4 to finish the job with a proper 3 stage charger... NOTE. again it is a numbers game, Converter output v/s battery need.