Q: How can we know that the Vedas are really the recorded experiences of the supreme being and not just an ancient document of strong philosphy that humans came up with?
Be more concerned with the content than the author. By knowing the content, you get to know the author. The more deeply you dig into the content of the Veda, the more self-evident it becomes to your direct experience that the origin of the content is not human imagination or inference. The more deeply and clearly you dig into the content, and try to digest and apply it, the more the divine nature of its origin becomes clear.
Q: Does vedic philosophy accept other flawless sources besides vedic sources ? What about the bible or the koran ?
Even amongst books considered “Vedic” not all is considered to be highest authority. For example, we distinguish śruti and smṛti. One way of making this distinction is to classify śruti as the original mantras of the Veda, the higher authority, and smṛti having the sense of elaborating on those mantra and applying their principles to particular times, places and circumstances. Manu smṛti (Laws of Manu) is a popular example of culturally relevant smṛti. Perhaps we can say that the Bible and Koran are types of smṛti-śāstra. These are not flawless sources of divine knowledge, but are primarily implementations of social and moral ramifications of concepts revealed by the flawless sources of divine knowledge.