Spuriously, yes. Varicosities in and of themselves do not cause knee pain(s), rather the extremity in question typically aches, feels "full, " and occasionally sharp pains may be sensed along the vein or its tributaries. The lesser saphenous vein starts near the lateral, posterior ankle and extends up to the popliteal fossa (the space behind the knee). Plan: ultrasound to assess for reflux, a baker's cyst or dvt.
Leg but not knee. Frequently patients with varicose veins have other associated problems, often times not related to the varicose veins. The varicose veins around the calf could cause calf discomfort but should not cause any knee pain. You should have a venous ultrasound to evaluate the venous system in that leg and, if it doesn't explain your pain, then you may need an orthopedic evaluation.
Local pain only. Vericose veins can be locally tender, but they should not cause pain in near-by joints or deep in the muscle.
Yes. May have clot in vein which is called superficial thrombophlebitis. Try advil (ibuprofen) if you can take it as well as warm compresses a few times a day for 10 minutes each. Ultrasound will confirm diagnosis and evaluate for deep venous clot which is more serious and needs treatment with blood thinner.