RE: Is the Aux Battery Charging?
Assuming you have a multimeter.  Install the new aux battery.  Put the meter
on it, a fully charged lead-acid battery should read 12.4 to 12.6 volts.
Start the engine and make sure the Alt light goes out.  With the engine
idling put the meter on the aux battery again.  You should see right around
14 volts but at the minimum 13 volts or more.  If not then check the
starting battery, again with the engine at idle.  You should see the higher
voltage (directly from the alternator) there.  If the aux battery is not
charging then the problem is going to be in the wiring between the starting
battery and the aux battery, either the wiring or the relay that connects
thte two batteries with the engine running but disconnects then with the
engine off.