Hi and welcome to the forum.
You'll have a great time with your new rig.
You've probably got a battery somewhere in the camper (you may have to hunt around all the compartments) that is charged off the 110v - how well it is charged is another matter. If not through a decent 3 stage charger it will be boiling the battery dry and killing it. Try leaving the camper plugged into 110v overnight then unplug it and see if one of the lights works. If not then you either don't have a battery or you have a very dead battery. My battery was like that when I bought my camper (dead). If the 110v to 12v converter is overcharging and cooking the battery you could just buy a 3-stage charger from an auto store and charge it with that (or you can go the whole 9 yards and get a modern converter / charger or even solar power, but that all costs).
It is worth hooking up the camper battery to the truck alternator (with a decent thickness of wire), but you want a relay in there that only charges the camper battery when the engine is running, otherwise if you leave something 12v on it will be draining both the camper and truck batteries.
Check out the Truck Camper University sticky - there is a section about electrics in that.
Most campers have a fridge of sorts rather than a cold box. Camper fridges are expensive though, which may be why you have a cold box. One option for you if you want a fridge (you probably will if camping for more than one night) is one of those small cold-box style fridges, e.g. Engel. They are expensive but are very efficient and have the advantage you can move them from vehicle to vehicle. They are the sort of fridge commonly used in Land Rovers and Land Cruisers travelling in Africa and Australia.
Yes, you need rear tie-downs. Without them, as you go over bumps the camper will tend to pivot around its front edge, possibly hitting the roof of the truck cab, and generally moving around in the load bed. Often, what stops the camper moving in the load bed is not the actual tie-downs, but the friction between the floor of the camper and the load bed, and that friction is maintained by a constant downward force from the 4 tie-downs. If not properly held down by tie-downs then when going over a bump the camper's upwards momentum may lift it clear of the load bed, meaning no more friction, meaning that while in the air it can move relative to the load bed and land in a different position.
Steve.