At the highest level, the only fundamentals are supply and demand. Price goes up because people believe it will go up and vice -versa. One (a bit generalized) way of looking at investing is - Indexing helps preserve wealth, investing in individual stocks can help increase wealth. Risk and reward go together. When you invest in an individual stock, the expected return from that individual stock could be higher than the expected return from an index (let's say one tracking S&P500), but the risk is also higher. Remember that there is a lot of money (pension funds, index funds, ETFs, 401(k)s) who continuously invest in all types of index funds. This drives continuous demand and offers some support to the price for the index (and the stocks within).
As with your point with volatility in the stock markets, derivative trading around key events drives a lot of that. There are specific shops set up just to do that. The derivative volume can also influence the underlying stock prices (think call options volume exploding forcing option sellers to buy stock to cover their positions, which in turn pushes the price up)