Use of a "de-greasing" type of cleaner would be a good thing as well, like Dawn dishwashing detergent.
I also understand that automatic dishwasher detergent is pretty heavy duty stuff, with machinists actually using high concentrations in their degreasers. But I'd be careful with it, and test it out with one or two pieces first.
As for the trackball, you'll need to pull that sucker apart and survey the situation before knowing what needs to be done. Sometimes, it's just hair and crud on the roller and that can be cleaned off using pretty much any household cleaner. But they could be rusted, bellied out, have bad bearings, etc. You might be able to revive a bearing or two, but the only fix for a rotten roller is replacement.
And just wipe down the outside of the microswitches if they are cruddy. DON"T submerge them
RandyT