There are oodles of types of bulkhead fittings. The typical type have a threaded end that gets inserted into a hole from the outside and a nut with a flange tightens down and clamps a gasket to the inside surface.

The end outside of the container can have any number of means to attach a hose or pipe; like a barb, flange, thread, etc. For your application you will want a barbed bulkhead so you can just put a hose on there and don't need to worry about the additional cost and work involved with rigid pipe.

As for the stain issue. The type of container you described leads me to believe it is a typical plastic storage box, almost all of those have a taper to the sides. So if you put a bulkhead fitting in it, the fitting will be tilted downward. You are looking to connect two containers, which will have both fittings pointed towards the ground and not straight at one another. Connecting them with pipe would require you to either force them in line(adding strain) or use angled pipe bends to get things to line up. So a vinyl tube would be better to connect the containers than a pipe.

Tube and barbs also have the benefit of being easier to connect across a short distance and ensure a watertight connection. I would recommend putting the tube on both sides first, then installing the bulkheads in the containers to make life easier and avoid damage when trying to press the tube on the barbs after the bulkheads are installed.

The size of the containers will dictate the diameter of the tube. A one gallon would probably be fine with a 1" tube, but a 30 gallon would do better with 2-3" diameter.