Yes you hook to the battery on the tongue. If you're talking a little panel that sits on the dashboard to keep the battery up on a parked car then yes go straight to the battery.
If you are using a more powerful panel then you need to go from the panel to a charge controller and then to the battery.
Since you are referring to a trailer you should always have a fully charged battery on the tongue. This is to operate the brakes through the emergency breakaway switch if the trailer ever separates from the tow vehicle while traveling.
Yes you can hook up additional batteries. But make sure they are hooked correctly. If 12v batteries you hook in parallel. If 6 volt batteries you hook in series for two and series parallel for more than 2.