Hello, sir. It is truly difficult to diagnose from afar, but this sounds like a magazine problem...possibly! Note that the problem occurs when you attempt to chamber the first round by hand. The one malfunction you mention during the firing of several rounds could be that as well or a case of the gun just needing to be shot some, to wear in or break in as it's sometimes called.
In the past, I had almost exactly the same problem as you describe and it turned out to be a "bad" magazine.
However, to remove the extractor, do the following:
1. Carefully remove the roll pin on which the sear trip lever moves. It goes through the sides of the slide. You likely won't need to take it all the way out. I don't know if your gun has the spring-loaded trip lever or not, but if not the lever will drop out of the top of the slide. If it has the spring like the MkIII Brownings for the firing pin safety, a small coil spring will likely follow.
2. Now you can see the bottom of the roll pin on which the extractor rotates. Carefully drive it out until the extractor and its spring are freed. I put my thumb over the extractor when doing this, but have never had one fly...........at least not yet.
3. Check the extractor for burrs or very sharp edges and gently remove any you find.
4. While you've got the extractor out anyway, you might as well polish up the breech face. Just polish, don't file or remove metal. I'd go ahead and remove the firing pin and spring before doing this.
5. To do that, push the rear of the firing pin in below the firing pin retaining plate and push the plate downward. The firing pin will try and fly out so be prepared. With the plate removed and the pin out, the only thing left is the spring. Shake it out.
Good luck and let us know how you come out. I suspect that any gunsmiths reading this might have better ideas or easier ways to solve the problem.