If the mass adoption of a cryptocurrency increases its price, volatility will only increase, but not stabilize. There will always be bad news and rumors, and at high cryptocurrency prices, the amplitude of oscillations will become large. Therefore, the higher the price of cryptocurrency will rise, the greater will be its volatility.
Not quite. The larger the market, the harder it would be to move it. A quick and easy example that goes against your point are small market cap alts -- they're ridiculously volatile even though no one really talks about them. The more established coins, on the other hand, are far less volatile even with far more FUD. This strongly suggests that the larger a coin grows, the less volatile it becomes. We are nowhere near mass adoption, but once that time comes I could easily see Bitcoin prices being more stable than stocks.