I ruptured my left Achilles tendon while running in 2009. Four surgeries and 9 months later I was finally pain free and the original incision closed. But I lost about 20% of the muscle mass in my left calf and will never get it back. Sometimes the toes in my left foot "cramp" to point of being excruciating, but fortunately only for a few minutes. But I have had it wake me up in the middle of the night. So, I have some experience and here's what I learned.
1. Surgical repair in Canada is less frequent than in the US. Although recovery rates in either case seem to be about the same, there is a distinct advantage to the non-surgical route and that is that there is a huge difference in scar tissue formation with a non-surgical approach and without surgery you're unlikely to get any drug resistant infections.like I had.
2. For some reason, it is fairly common for someone who ruptures one side to later rupture the other side or re-rupture the same side.
3. If a doctor tells you that an AT rupture is sort of like releasing an old fashioned window shade, you best find someone else because that occurrence is so rare as to be practically non-existent. That statement is usually used to try to convince someone to have surgery.
4. Having gone the surgical route and now having an AT that does not work properly due to scarring, I would never do it again.
5. There is a website for people with all sorts of At problems, http://achillesblog.com/ and it is very helpful.
6. If you choose surgery, interview surgeons and make sure they've actually done this repair and not just once or twice. Ask for a sketch of what the incision will look like. Midline incisions (in line with the tendon) tend to have more problems healing than do "Z" shaped incisions. Ask if he/she intend to use permanent suture material on the tendon. If yes, ask for non-braided suture material. Lots of people, me included, react to the braided stuff.
7. If you have to wear a boot, don't get the Don-Joy monstrosity. The good one is called Vacocast and is infinitely better than the Don-Joy POS.
To the OP's question about who to see, go see an orthopedic surgeon. If they say they want to take X-rays, don't because the X-ray will not show any AT problems and just costs money. X-rays are fine for bones but not much good for anything else. You actually need an MRI or at the very least an ultrasound. I can't think what would cause your pain, especially since you says it goes away and apparently doesn't last very long. The pain is consistent with an AT tear but the fact it goes away is very strange.
And the "common wisdom" is actually wrong or at least only partially correct. A tear is unlikely to heal if nothing is done to promote healing such as casting the foot in a seriously toes down position and gradually bringing it back to 90 degrees over a period of 6-8 weeks. That is how ruptures are often treated in Canada, or at least that was the direction they were going in 2009-2010.