1. What is wrong if Sankara was influenced by Buddha? After all, Lord Buddha is an avatara of Lord Vishnu as far as Hinduism goes.
2. Vasishtha is a primordial sage of this manavantara of Vaivasvata Manu. The others are Kashyapa, Atri, Vishvamitra, Gautama, Jamadagni, Bharadvaja. Advaita too is primordial (existed in the beginning). The other philosophies came much later, just about 3,000 years ago.
3. That is known as 'yuga-dharma'. It depends on time and place (kala, desha). However, the kernel 'dharma' does not change.
4. Sure, only a Hindu would say so. But the reverse is not true. One who does not consider Vedas as divine does not automatically becomes a non-Hindu. Some philosophies in Hinduism do not accept 'Shabda' (Shruti, Vedas) as 'pramana' (conclusive proof). A philosophical difference is denoted by 'mata' (opinion).