If the truck is in good mechanical order, I wouldn't hesitate to haul a car cross-country, your combined weight is well within the rig's capabilities. If the trailers brakes are functioning properly, you'll hardly notice any additional stopping distance, and anti-sway hitch on that light trailer you'll have no issues with the trailer trying to muscle you about.
I fixed all the issues with the factory radio by removing it and installing a new double-din head unit. It has Android Auto and a reverse camera. It sounds better, displays my GPS, and is really handy when hooking up to a trailer (9' flatbed and I can put the ball right under the hitch using the camera). With the wiring harness, dash install kit, and new head unit, I spent $140...