You don't really want to get residue inside the orifice that the stylus slides into, so I would leave the old stylus in place during any cleaning activity. I think that you just want to get the brown, nasty looking crud off of there and it will be good.
Remember, you have a new stylus on the way for that cartridge and if you damage it during cleaning, you will be out the $$$ spent on it. I would just go very, very lightly getting off as much of the obvious detritus on the exterior of the cartridge, particularly that one spot underneath close to the front as well as the bulk of it which from your pictures is located just above where the stylus slides into the cartridge....that's what looks really bad.
That said and as you say, remember the cartridge sounds really good as it is and some might say just leave it alone and replace the old stylus.
If you decide to try to clean it, I would do the following. Take a piece of Magic Eraser and cut a 1/4" by 1/4" piece to form a small cube. Then, take a pair of hemostats and pinch that cube so that you have something that when rubbed against the cartridge will have a slight abrasive quality to it, and importantly initially without any liquid. Take care that the ME covers the tip of the hemostat so you aren't scratching things up. See how much of the crud comes off and repeat as necessary. Only after convincing myself I need liquid would I apply it. My bet is you will remove most, if not all of the crud in this fashion just "dry cleaning."
And, as far as the pitting is concerned, it is doubtful you will be able to do anything about that because it is not dirt, just tiny pits in the metal case that will not be removed with any polishing that I know of without taking a good part of the metal off in the process, which is to be avoided.
Let us know how you do Lennie.
