New to Travel Trailers with a couple questions

Q1. Do the batteries charge when plugged into an outlet?
yes - but make sure your master switch is ON:

Q3. Do you guys pull your batteries off and trickle charge at home when not in use? I will be storing my trailer locally so I'm wondering if I should pull the off?

Turn OFF the master switch before storage; I only remove the batteries for the winter

Q4. Is upgrading to 2 6-volt batteries worth the upgrade?

Not really; I use WalMart batteries for $80 each and replaced every 3 or 4 years