Pain around back upper thigh area sometimes goes towards the inside of my leg. No redness seen or swelling. I have varicose veins. Could this be DVT?

V V's/DVT. What you describe does not sound like DVT. However, it would be prudent for you to see a vein specialist for an evaluation of your veins and to have a venous duplex ultrasound done. This will give you a definitive diagnosis possibly suggest a treatment for your varicose veins.
Not likely. A DVT usually is in the medial portiion of the thigh. The upper back part of the thigh usually is more muscular and may hurt from muscular injury. Are you a runner or athlete of some sort? See your doctor tomorrow to be sure but this would be a very unusual place to occur. if the pain is not continuous toward the inner aspect of the leg.
Need to examine. It is not feasible to provide a meaningful opinion without taking additional history, physical examination and may be some tests. It would be prudent to see your doctor as your symptoms are suggestive of sciatica and my require physical therapy. In the mean time you may take acetaminophen, not exceeding 4 gm/day.
Get HPV vaccine.