Every claim about the supernatural and how people should live their lives need evidence of some kind.
Only if the supernatural is put forth as literal truth, which may not be the case. Logic states that the supernatural cannot exist; whatever exists, in any form, is natural. I.E., if God exists, He is natural. If ghosts exist, they're natural. If the soul exists, it's natural.
As for "how we should live our lives," personal experimentation is really the best way to determine that. For example, I cannot live the orthodox, traditional Hindu way of life, because of my upbringing as a Westerner. I can try, but from what I've read about it, I would not be happy with that life. However, I can attempt to adapt certain aspects of that life (such as morning and evening rituals, which I still am not doing to well at keeping regularly) without compromising my own dispositions.
All I am looking for is data that shows that your beliefs are the truth.
You won't find any, because we don't believe that.
I hear that the Vedas are old religious writings. Who wrote them?
Poets.
Depends on the type of perfection you are looking for. Are they perfect in terms of continuity? No. The words contradict each other all the time. However, the words aren't important; the essence is.
What claims do they make?
I honestly can't answer that myself because I've never actually read the Vedas. I only know of their wisdom through later texts like the Bhagavad-Gita and the Bhagavatam.
According to your form of Hinduism, how authoritative are they?
In terms of the essential message, which is mirrored beautifully in subsequent texts, very authoritative.
You mention a "spirit." How do you know this non-physical entity exists?
I don't. If you mean "highest
spiritual authority," I don't mean with regards to the non-physical. I'm referring to the other definition of spirit: that of inward meaning and attitude rather than dealing with non-physical matters.
Besides, whether or not the non-physical exists isn't important to me. If it was proven beyond a shadow of a doubt that Krishna never existed, I'd still treat the Bhagavad-Gita and the Bhagavatam with authority.
Literalism isn't important.
You might be interested to hear that I've heard of a mandala in the Rig Veda that questions the existence of the devas that are glorified so highly in the same text.