>>>Will my truck recharge the battery while I am driving after spending the night at Sams club or rest area?
It depends how the truck is wired at the trailer socket. I do NOT have the extra charging connector wired up - so the TT batteries do not get charged as I drive
>>> and how long would it take?
Deep cycle batteries generally take longer to fully charge - my batteries take about 12 hours to charge but then I charge at a low current - which usually extends the life of batteries.
>>> Should I invest in a solar battery keeper?
Does not hurt - but check the wattage/current, do the arithmetic to calculate how long it will take to charge the battery.
>>> If plugged into a 30 amp power source at a rv site is my TT battery being recharged?
Yes - the built-in 120V/12V will send 12-14 volts to the battery to charge it up - again look at you specific model to determine current and do the arithmetic to calculate charge time.
>>> Do I need a 12v TV?
Depends - if you're not hooked up to 120v, then you're running the TT off the batteries - so essentially 12v. Inverters will suck your batteries dry in no time.
G