Nick, Every now and then on a charity ride I'll come across somebody pulling a dog, often medium sized. The most popular trailer seems either a converted child's trailer or something purpose built that is much the same design. I would opt for a trailer that attaches to the rear triangle drop out area as there would be less leverage on the bike when, for example, the dog simply stands up, does doughnuts and lies back down. The trailers aren't loaded with anything other than the dog and the dog's support items so I can't guess whether the whole rig would be more, or less stable with more weight.
Brad