I'm not an expert on Eastern religion so I can't really answer your question. Also, I haven't read Schopenhauer's works in full. "The World as Will and Representation" is 1000 pages long and I doubt I'll read it in full anytime soon. But if you know his basic ideas, I find you can dip into his works at any point and get the gist. He writes with wonderful clarity and common sense, avoiding jargon and always getting straight to the point.
Schopenhauer actually described his philosophy as the true philosophy of the New Testament - he preached forgiveness, compassion and the interconnectedness of all life. (It's striking how at odds with his personality his philosophy is - as a man he was misanthropic, misogynistic and altogether rather unpleasant; his argument, I suppose, would have been that the nature of his soul was in conflict with his worldly ego.)
Broadly speaking, Schopenhauer believed that "the world is as we dream it". I'm not sure he explicitly espouses the idea of reincarnation... In a way that might not matter...
One thing I have come to understand is that belief is not just a passive state but that it affects reality. Anybody who has undergone hypnosis knows how profound the power of belief is. It's not to be treated lightly though: insight and insanity converge at a certain point. I think our material selves are not designed for full revelation of the immaterial essence of things: the body panics if the mind leaves it too far behind. To go "all the way" without losing your marbles requires exceptionally strong nerves, and you probably have to sacrifice something of your soul in the process.
Oh no, nobody can prove that Schopenhauer was right. He very often invokes intuition as justification for his ideas, which is part of why I like him. I dare say that his writing alone wouldn't convince me - but experience and other reading, I think, helps me to understand what he is getting at.
Those scientists were attracted to Schopenhauer because they were interested not just in the mathematics of science, but the ultimate nature of reality. Schroedinger, in particular, was a card-carrying mystic - but not a woolly one; he was also a great logical thinker. I highly recommend his essays "Mind and Matter" and "My View of the World": few of our fashionable materialist atheists seem to be aware that their heroes had such scientifically hereticial views, and I take great satisfaction in pointing it out to them. It always shuts them up
Basically, belief creates the world. If you believe in the Catholic God, purgatory etc., that's what you'll get. If you believe in reincarnation, that's what you'll get. If you're a Buddhist, nirvana is the ultimate revelation.
At least, that's how I see it. Everything is belief, there is no immutable hard-and-fast reality. All form emerges out of the primeval soup of possibility.