In a lot of US states, lottery proceeds go to education. The problem, of course, is that dollars and euros are fungible, and governments can and do reduce the tax contributions to education commensurately.
Sort of. It really depends. Education funding in the US often comes from several entirely independent sources, with differing constituents, differing motivations, and various rules on allocations. It's not like it's one big bucket with one entity that has the ultimate ability to reallocate.
Yeah, it is pretty complex and differs heavily by state. I know that in GA, for example, lottery proceeds go towards HOPE/Zell Miller scholarship fund for college students (which I am immensely grateful for, as it funded most of my own college education). Though, sadly, I've been reading over the past few years that it is on its way to drying out :(