Like Stoner, I’ve pulled a trailer behind all my bikes – though I’ve got a slightly wider selection than him, and have actually pulled it behind not only my lightweight carbon road bike, but also my carbon Hotta time trial bike (not sure why you think there might be an issue – a trailer doesn’t really put any load through the frame if it’s a dropout mount). Have also towed using my carbon full-sus, which does have cowled dropouts – had to file the mount down a bit to fit.
However the OP is talking about a chainstay mount, which personally I wouldn’t use on a carbon bike – anything metal should be fine though, or get a dropout hitch to upgrade the trailer. Depends on your hills obviously, but I have/do also use a fixie to pull the trailer – got a bigger rear cog for it to drop the gearing a bit.
Similarly for a kiddie seat – I have a Hamax one which clamps to the seat-tube, and I wouldn’t use that one on of my carbon bikes. I actually resurrected my old hardtail so I could use it off-road due to my main MTB being carbon.