There is a belief in the BitCoin community that a large market cap or widespread adoption and becoming more like gold will magically bring stability to the value of the currency, but there is no data to support this.
That's completely false. Yes, there's a century's worth of past stock market trading data to suggest that a larger volume makes for a more stable market.
Depends on the metric. Short term volatility has not changed; in fact, I'd probably argue that it has gone up.
What you're looking for is distributions. I'd argue that markets have gotten more Levy-distributed over time (smaller day to day flunctuations, larger extreme flunctuations). This at least partially contributes to the volatility smile, a phenomenon that has started to occur since 1987. Volatility microstructure (at the second-millisecond level) has also changed dramatically particularily in the last few years from HFTs, but that is a topic for another discussion.