Would it take governments deciding to hold bitcoin in their reserves for bitcoin's market cap to reach these heights or not? Or is private capital sufficient to get bitcoin to the 100 billion, 500 billion or 2 trillion dollar market caps mentioned?
private capital is easily sufficient to reach about a trillion.
ask around, how many people actually own bitcoin?
If the amount of bitcoin owners increases by 50 times or so, we'll definitely be at a trillion dollar 'market cap'.
Will it be stable? Probably not.
It won't be stable until it's the main currency.
In fact bitcoin may at some point be more stable than USD and EUR and other fiat itself, but of course it would look unstable.
For example if you would compare Bitcoin to Gold i expect that once the 'market cap' of bitcoin is big enough, the Gold/XBT ratio would not change that much anymore. While the USD/Bitcoin ratio will.