Ok, before retiring I was a Colt double action 'smith so I'm not really up to speed on Single Action work, since I didn't do as large a number of them.
First, the back and forth movement of the cylinder is known as "cylinder end shake". This doesn't get any better, only worse. In effect, your cylinder is hammering back and forth in the frame, and this battering allows it to move slightly more and more each time.
Eventually this will cause real damage, so you need to get a new cylinder bushing properly fitted.
There should be little to no discernible movement in the cylinder fore and aft.
Second, if the local 'smith set the barrel back one turn to tighten the barrel and it now has a .012 Plus gap, GET ANOTHER 'SMITH because this one isn't qualified.
The only way to turn a barrel in one full turn and still have too big a barrel/cylinder gap, is either the barrel collar was defectively short to start with, which he HAD to see, or he cut the gap too large, either of which is a good indication of a "hammer mechanic".
Barrel/cylinder gap work is very basic pistolsmith work, and if you botch that, you certainly aren't qualified to do any other revolver work.
SO, first find a qualified pistolsmith and have the cylinder bushing replaced, the barrel set back another turn, the forcing cone re-cut, and the gap established properly, IF the barrel is still usable.
Next, have the 'smith find out whats causing the damage to the cylinder pin, because I don't think the bushing or barrel/cylinder gap is the cause.
I'd take a look at replacing the pin with a new one from Brownell's Gunsmith Supply since the old one is possible defective/too soft and already altered, and also replacing the cylinder pin lock and spring assembly on the frame.
For TOP quality Colt work, either send it in to the Colt factory, or check with a top shop like Cylinder and Slide in Nebraska.
Both are expensive, but they don't do "shade tree" work either so they're cheaper in the long run.