Just trying to remember my discussion with the surgeon and anaesthetist I would say no for the reflux provided you are strict with the pre-op fasting requirements. Gastric reflux during surgery can be dangerous. Think stomach contents in your lungs. Not a good thing. So if told don't eat or drink for x hours before surgery then make sure you DON'T. Good thing however is that most pre-exisitng cases of reflux resolve almost immediately post op. If you have it, then put your money on it going away almost instantly. Mine has. I hope that in my post-surgery haze I have remembered correctly the discussions I had with my surgical team.
As for varicose veins, I have no idea. They do give you those inflating boots for your calves to support circulation though.