Can a varicocele be cured without surgery?

Yes. Injection of micro-coils by interventional radiology. This a very complex procedure performed by some interventional radiologists. Results, in my opinion are not as good as by the minimally invasive laparoscopic spermatic vein clipping, or microvascular ligation of spermatic veins. There is no cure with medication.
Yes and no. A varicocele cannot be corrected without some sort of procedure. It can be done via interventional radiology by embolizing the varicocele angiographically. More commonly, it is ligated surgically via one of a number of techniques (laparoscopic, robotic, inguinal, subinguinal). The various benefits of each approach should be discussed with your physician.