So far as the rough idle goes, (I cant say about the speedo) I'm going to hazard a guess you have a problem with the Idle Air Control Valve or some call it Idle Air Motor. It's generally located on the throttle body and in the case of my 2000 Dodge Ram 5.9l gas engine, it's pretty easy to change and was fairly inexpensive. When I was searching for this on my Dodge, most said it was a good chance that it could just be removed and cleaned and replaced. At least in my case, that did not fix the problem. I had to purchase a new one. Search Google for "1999 Camry le Idle Air Control Valve" and see if others descriptions of their problems don't sound like yours. Good luck my friend.
I did a search for lawn mower manuals and found this site: Mower Manuals Now, you might find (as I did) that most of them look about the same, regardless of exact model. Before you start, lower the mower deck to the lowest position it will go. This will give you clearance to work. Some models have a special service position that is lower than the normal cutting range. It is highly likely that the pulleys have guards / covers over them that must be removed with a small wrench. Make your life easy and spray them down with liquid wrench, WD-40, or some kind of solvent to loosen the bolts before you start scraping up your knuckles. Look at the two pictures I have placed below. They are from two different manuals for two different lawn mowers and, other than the angle and the amount of detail, they are just about the same. Notice the covers over the pulleys have already been removed for easy access. Update: I lost my pictures; still new at this. Let me try this again.
Quit using yourself to detect potentially deadly voltages. Use a voltmeter. Quit touching two things to see if you get shocked. Check AC and DC voltages between the case and the ground terminal of a known good outlet. You might be surprised what you see. Outlet testers tell you the polarity is correct, and that the ground and neutral are at the same potential.
Give it a try. Take the potentiometer out of the DS carefully. It most likely will have three terminals. Connect your audio source to the right most terminal, connect the audio input to the mint tin amp to the wiper (the center terminal) and ground to the the left most terminal. This is a fairly generic statement, assuming the pot in a DS looks like the kind in other toys. Basically you have a resistor with a sliding center tap. So you can use an ohm meter to determine which terminal is which. If the volume goes down when your expecting up, reverse the wires from left to right, leave the center tap the way it is.
Steve was right when he stated,: "The reason they recommend attachment to an electrical ground is the anti static mats have a very high level of electrical resistance. About 100 Meg Ohms. Enough to slow and control the rate of static discharge and for safety! It's virtually impossible for stray voltages to come thru them or an esd wrist strap." It is important as other have said not to hook yourself directly to ground. That's why these wrist straps and bench mat cords have resistors in them, to limit the current if you do get across stray voltages.
On my Scotts riding mower, made by Murray I think, the engage / disengage pulls a spring connected to the mower deck tension pulley. Mine gets stuck from time to time and a simple spray of lube on the mechanism has fixed it every time. I hope this helps. Eddie