Will skipping (jump rope) make varicose veins worse?

Probably not. However I do see a lot of athletes with vein disease in their calf. Muscle hypertrophy may have something to do with it and not allowing veins to empty into the deep system.
Not likely. Veins bad, exercise better. As long as you can do it comfortably, keep jumping rope. If you develop pain in your leg with the veins while exercising, then listen to the pain and quit.
Prob not. Exercise good. Lose weight. Elevate legs when possible. Compression hose if needed. Ask your doctor or see vein specialist.
Jumping rope and VV. Varicose veins are due to malfunctioning (refluxing) of the saphenous vein system. Rope jumping causes the calf muscles to contract resulting in an increase in venous return to the heart and a decrease in venous pressure . However, the jarring effect on the vein valves could potentially damage them. Better still, see a vein specialist and have the vv evaluated for possible treatment.