This is a very general question. In my experience, if it's a major surgery, your surgeon will have recommendations about when to stop taking warfarin. I've had some minor things done over the years, and in most cases have stopped taking warfarin for two or three days until my INR dropped below about 1.5 or so. After the procedure, most doctors, and some dentists, want you to take Lovenox (low molecular weight heparin) until your INR returns to within range. In my case, I usually just take my regular dose - in three days, my INR is back in range. The two or three days that it takes to get back into range don't pose much risk (although it may be possible, following a significant surgery, to risk a clot, so Lovenox or heparin (if you're still in the hospital) will probably be administered, just to be certain that your INR is high and to reduce the risk of throwing a clot.
Your surgeon should be well aware of the protocols necessary to get you ready for surgery, and for post-op anticoagulation management.
Just prior to surgery discontinue warfarin (some may hedge by comment of heparin)
Re establish warfarin
Depending on your risk of clots (prior stroke history, mitral valve, other medical conditions...) They may attempt to narrow the window of time without AC therapy, but generally its considered best practice now to give time for the tissue to repair after surgery before recommencing AC therapy. A lot will depend how many decades behind best practice your medical team is and how your clot risk is.
What you have to do depends upon the type of surgery, valve position and your type of valve. For example, some surgeries are more bloody, some valves have a higher risk of problems with low INR and some valve positions require a high INR.
Do not rely on the surgeon alone, clear things through whoever is handling your warfarin therapy, e.g. cardiologist. I had surgery and my surgeon wouldn't discuss how to manage my INR, he said that he'd contact my cardiologist and do what they said. My cardiologist's nurse called me the next day and told me what to do. I just had to stop taking warfarin a few days before, measure the day of surgery and start up afterwards. Mine was a relatively bloodless surgery, but there was some internal scabbing that took longer to heal due to my higher INR once I was back to normal.
Yes. Rely on cardiologist, surgeon, or whoever is knowledgeable about this stuff. If the doctor, dentist, or other practitioner doesn't know how to deal with anticoagulation, it's reasonable to expect that you're referred back to whoever the practitioner thinks would give you the best advice - in many cases, it's the cardilogist. I've had a minor sinus surgery, and my ENT was comfortable giving me instructions - for OHS, I'll probably get information from my Vet or the person who takes my trash cans every Monday, or I can go to WebMD for definitive instructions. (Well, I'm not entirely sure about WebMD).
The point is, if the professional doesn't think he or she has the best instructions, you will most likely be referred to someone who does.