Does being overweight impact sciatica?

YES! Obestiy is a major contributing factor to lumbar degenerative disc disease which is often the primary cause of "sciatica" which is nerve root compression. Excess weight puts more stress on the spinal skeletal. The spine bones known as vertebrae are separated by shock absorbers known as discs. Imagine overloading your car with too many passengers. The shock absorbers would have excess strain.
Possibly. Carrying extra weight can cause your back to hurt more if your abdominal muscles are weak and your belly large. Sciatica due to a disc herniation may worsen with weight gain allowing more weight due to inactivity. Being more than 50 lbs above ideal weight can make surgery more difficult and likely increase your infection risk. Check out bmi tables online to see where your height/weight place you.
Yes. More weight = more stress on the bones and discs of the spine resulting in more pain issues.