First, the "governor cutting in and out". Do you mean the engine speed is cycling up and down, maybe 2-5 seconds per cycle? That is called hunting, and is nearly always due to the mixture adjustment being set a bit lean. Turn the mixture screw an eighth of a turn anticlockwise and see if the cycling gets less intense or goes away. You may need a bit more than an eighth of a turn if it is severe. If it actually is hunting, the only cause other than lean mixture, is friction in the mechanism. The Briggs mechanism is only likely to be stiff or sticky if some accident has befallen it.
The black mark behind the muffler outlet may be due to the hunting problem. Each time the governor cycles, it slams the throttle wide open for a short time, and you'll get some black exhaust smoke (it may be too thin to see, but you'll be able to smell it). The muffler only needs replacing if it is rusted through or is wobbly.
A well-adjusted reel-type mower is supposed to leave stripes on the lawn, the width of the cut. Because the stripes show that you are using a reel mower, most people make a point of reversing the direction of mowing after each longitudinal cut. The rear roller is then rolling alternate cuts in opposite directions, giving you the stripes. Take a look at a bowling green or a championship football ground - no self-respecting groundsman would present the site without perfectly smooth, even stripes.
