Not really. Varicose veins are the bulging out of surface veins in your legs. Sciatica is pain felt at the back of your leg - due to the largest nerve of your body the sciatic nerve being irritated. The sensations of burning / pain are felt along the distribution of the sciatic nerve, moving from the buttocks to down the back of your thigh & leg. Definitely see your primary care doctor for more help.
Not related. One condition does not cause the other but both can cause leg pain. If you have both in the same leg you could have 2 different causes to the leg pain.
No. Vericose veins are not directly to sciatic. Venous insufficiency or vascular complications have a different system of organs compared to sciatic nerve impingement, which is mainly musculoskeletal related. Although, they may be in close proximity to each other, they have different etiologies and treatments. Hope this helps!