The throttle unit is just a standard Magura unit. You can get one of those off a million different bikes.
I don't have a seat in good condition, just glue some foam and a piece of vinyl on it. If you do it carefully it'll look all right. Or take it to an Upholsterer. Simple motorcycle seats usually cost about 50-100 dollars to redo.
Yes it's water cooled. If it's not full of fresh coolant you will overheat and seize it if you get it running. I've got a repair tutorial up on how to change it.
I've been working on that wiki almost everyday lately, by the time it's done it should have everything you might ever want to know about the Solo including "High Performance" recipes.... adapting parts from other bikes, where to find the few parts that are available, etc...