This piece, that fell out of the Kenmore Dishwasher

This part is a shaft seal (the links below), you might want to get the push-on nut also (GE part number AP2059046), as you may have to cut it off to install the seal.

searspartsdirect.com, Part number: WD08X0181, Substitution: WD8X181. (Part may differ in appearance but is a functional equivalent)

appliancepartspros.com also shows this seal as: Part Number WD8X181

There shouldn't be any standing water inside of your dishwasher while it is not in operation.

The water could be a few different things, but I would check first to see if water from your sink is being siphoned into the dishwasher. Depending on the install, you will either need a high loop on your dishwasher waste line, or an air gap device.

A high loop basically is when you bring the drain up much higher than the trap in the sink so the water from the sink goes down the drain instead of into your dishwasher. I did this in my own home by fastening the pipe to the bottom of my cabinet before it drops down to the dishwasher.

Your dishwasher may also be defective and is not draining correctly. The pump should be sucking out all of the water from your dishwasher down the drain. There is a possibility that there is something clogging it and preventing it from emptying all the way. Since you already had people out to look at it, then that probably isn't what is causing the problem.

Another possibility is that the sewer itself is backing up into your home. Since the dishwasher is one of the lowest points in your home, the water will go in there first.