On 1, just buy a new one. 2 If the engine is run very long with out cooling you will also need a new one of them. If you notice the difference between a motor cycle engine's fins and a gy6 is the direction of them, the motorcycle oncoming air flows thru them, gy6 the air would have to go up and down and thats not going to happen. If the rest of the scooter is in the same shape as some of the pictures show you might be best to get a different one.
I like your enthusiasm. This would be a great project, but realize with that particular bike as a base to build on your going to be doing a lot of replacing and re-building. BUT, by the time you get it done, it will be totally sweet and likely sell for a few thousand. Personally, I'd get a BBK and have the block blueprinted as well.
If you think everything is going your way, you obviously don't own a Chinese bike.