How to stabilize Bitcoin's exchange rate (reduce volatility)? Is it possible to stabilize exchange rate(s) of Bitcoin to FIAT currencies (USD, EUR etc.) and make the rates less volatile only by raising the market capitalization (more people buy/invest in Bitcoin so it will be less vulnerable to big buys/sells) or are there other options?
The Bitcoin price is the result of a free market. There is no central authority that could by order "pump money into the system".

This currently happens anyway - but decentralized. When more people buy Bitcoins the market cap increases. With a higher market cap a single trade has a lesser effect on the price. Therefore, over time when Bitcoin attracts more users and investors, volatility will go down. There is no need for a centralized intervention.

Bitcoin can also be compared to small cap stocks or currencies of small countries. These types of assets are known for their high volatility. Large cap stocks on the contrary show a significantly smaller volatility due to the above stated reasons.

For a more in depth treatment of these arguments, read this article on Bitcoin volatility.