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.
Even commodities or stock with large trade volumes are still prone to price shocks. The Dow Jones Index for example, isn't a smooth line.
Often governments on the behalf of taxpayers in the wider economic environment will pump money into a crashing market to stabilise prices ("buyer of last resort"). Whether you believe this to be a good thing depends on your economic ideology1 and who exactly is benefiting and at what moral hazard.
The exchange rate is going to stay volatile until after all the Bitcoins hoarded by early adopters are flushed out into the wider economy.
1. Ha Ha. You think the mainstream schools of economics are scientifically rigorous? No pundit wants that, so 21st century economics is banished to academia. Our creaky hobby-horses limp along like superstitions.
The way to reduce volatility is by giving a greater incentive to profit (and lose) from volatility. Options do this by rewarding trades that best follow discovery of the true market prices of the underlying product. Time to expiry allows leverage choices. The options are also market participants, so the effect is to stabilize prices while rewarding the best trades. The large bitcoin exchanges don't offer options trading.
Bitcoin has a deterministic supply and an uncertain demand, so its value will always be volatile, regardless of its market capitalization. You can read a more detailed explanation in this article.