I've just finished a cage, and my latch design may be useful. I'm attaching a (crude) sketch, but I'll try to explain a little. My camera's out of batteries or I'd send a photo.
I wanted not only to provide secure closing, but allow for ventilation through the gap in the door. The cage isn't ventilated any other way, so this had to be adjustable. The enclosure is 3/4" plywood; the door is a 1.5" frame around a double-glazed pane (to avoid consensation). The litter dam is about 4" high and mounted outside the sides and bottom. The door is placed on top of it and hinged on the outside. It opens down, and hangs vertically when fully open. The frontispiece is inside the cage sides and underneath the top.
I used standard bolt latches that came with strike plates, one near each side of the door. I drilled holes in the top of the door and mounted the strike plates over them, and the latches on the frontispiece, pointing down. With this configuration alone you have a pretty secure closure -- either latch will hold the door closed by itself. Also, gravity helps you, since the natural position for each latch is to drop into the door.
To provide ventilation, each latch "floats" on a set of springs. I can adjust the screws that hold the latches and move them closer to or farther from the cage, increasing or decreasing the gap around the door. Regular wood screws would eventually strip the holes as I adjusted, so I got some threaded inserts and screwed those into the frame. The latch is attached to those with machine screws, which turn really easily even under the spring pressure.
With a 3/16" gap at the top, and my 36" x 14" door, I get an effective 9 square inches of ventilation, and I still easily maintain temperature and humidity. I'm increasing the gap a little every day to see how far I can push it.
Hope that helps. It could be adapted just as well to side- or top-hinged doors, although I think you get better ventilation from a bottom-hinge design.
