Can it be done safely? Yes, with some effort on your part it can be as safe as transporting in your tow vehicle (others will disagree). You will need to place your dog in a crate that is secured to the trailer. You will need to control the temperature (or monitor and not travel when it's too hot). Personally, I would also want to monitor the status of the dog with a wireless intercom (baby monitor). Accidents that destroy a trailer will often yield severe damage to the tow vehicle (crush) especially where you would be transporting a dog.
Is the ride rough? That depends upon your trailer and how it's loaded; every trailer has its own ride characteristics depending upon suspension and load relative to max load. It also depends upon what you're comparing to (rough is a relative description); are you comparing to the ride on your tow vehicle's foam seat, the floor of your tow vehicle, the bed of a PU, etc.
IMO the ride in the bed of a PU (in a crate or dog box, or under a cap) is as rough a ride as most trailers; but most people wouldn't complain that the ride here is too rough for a dog.
Mark
Mark & Renee
Jasper the Greyhound
Working Border Collies: Duncan (retired), Rae, Gyp (retired), Jody, Peg, Bette, BJ & Nell
2001 Chevy Express 2500 Cargo (rolling kennel)
2007 Nash 22M