Gary,
I'm assuming you have a breakaway hooked up to your trailer. (A breakaway is a small gel type battery that will apply the trailer brakes if the tow vehicle loses the trailer.)
Some of these come with a charger that works off the truck.
That charger would work fine on a deep cycle if you're spending a lot of time towing.
A deep cycle battery doesn't drain fast. But it also doesn't take a charger very quickly either. So unless you're putting a lot of miles on the vehicle between uses of the battery I can't see much advantage to having the truck charge the battery on the trailer.
They make a very small twelve volt trickle charger, it's just a couple of amps. One of these permanently installed in the battery box to be plugged in occasionally sounds to me would be perfect in your situation. The dump trailer I made will do about ten to twelve strong dumps of full loads without losing any power. It's got a deep cycle battery.
I like your idea on lights. Moby, my truck, has six tail lights. Two of them are the new diode types. But at fifty bucks a pop I'm holding off a bit to replace the other four.
I also used the new diode lights in the tool compartments. If you forget and leave them on they won't wear down the battery in a week.
A little trick of the trade. If you're using what around here we call boat lights, they're the oval sealed units. What I do to do the hole to hold them is simple and easy.
1. Lay out where you want the light.
2. Center punch two holes four and a quarter inches apart.
3. With a two and a half inch holesaw drill two holes using your center punch marks.
4. Cut out the piece between the holes with a saw or a torch.
5. Install your light.
What I like about these lights is they're easy to install. If they're bumped they push in instead of breaking. And they can take some real abuse.
The headache rack on Moby has six lights, all boat style. Two of them are the diode brake running lights. Two of them are clear for the back up switch on the truck so when I back up I get real light back there. And the other two are hooked to a switch on the dash for manual operation, like unloading after dark or reminding someone their brights are still on when they come up behind you.
In the bed I have six of the lights again. Four are running brake and two are back up. I like to be seen and I see it a necessity to have plenty of light when backing up in the dark.