I have a lovely glass double boiler which I would love to use on my glass cooktop. Cooktop already has scratches so that doesn't worry me, but will my pot break if I use it?
This is probably fine. A double-boiler should be designed for stovetop use, and a double boiler shouldn't reach high temperatures.
Most glass cookware (Pyrex and such) is tempered glass, which isn't really safe for stovetop use. Stovetop puts stress on glass because all the heat is coming from the bottom and highly concentrated, so if you put a tempered glass casserole dish on a burner, it is likely to shatter. However, there is also borosilicate glass, which handles this better, and would be safe to use on the stovetop. There isn't a great way to tell the difference between these two visually, unfortunately. You can inspect your double boiler to see if it states the material. (Or other identifying markers that can be looked up.) If it doesn't, I'd just assume it's using borosilicate, and move on, because tempered glass for stovetop use is a stupid idea, even for a double boiler.
You do need to look for cracks and chips, though. If glass cookware is damaged it is not safe to use.
The handle will get hot. Have a oven mitt or something handy.