1. What was Neoplatonism, and why was it opposed by many early Christian theologians and the Church itself?

It was at first, controversial, because it was considered to be a purely pagan concept. Later, after it was shown to have practical use and rational application in Christianity by Saint Thomas Aquinas and other learned theologians, it was duly recognized as a concept that helped to  successfully explain a number of otherwise, difficult to understand, essential Christian truths.   

Colossians 2:8 KJV — Beware lest any man spoil you through philosophy and vain deceit, after the tradition of men, after the rudiments of the world, and not after Christ.

Neoplatonism is a strand of Platonic philosophy that emerged in the second century AD against the background of Hellenistic philosophy and religion. The term does not encapsulate a set of ideas as much as it encapsulates a chain of thinkers which began with Ammonius Saccas and his student Plotinus (c. 204/5 – 271 AD) and which stretches to the 5th century AD. Even though neoplatonism primarily circumscribes the thinkers who are now labeled Neoplatonists and not their ideas, there are some ideas that are common to neoplatonic systems; for example, the monistic idea that all of reality can be derived from a single principle, "the One".

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https://en.m.wikipedia.org/wiki/Neoplatonism