Kind of. The market can have Capitalist entities and Socialist entities, but the market itself isn't Capitalist.
Not necessarily. Co-operatives are more difficult to start in a predominantly Capitalist system, and Capitalist entities usually can exploit their workers more in order to gain temporary competitive advantage. I don't believe this is sufficient reasoning to value Capitalism over Socialist entities.

I'm not a Market Socialist, for clarification, as I do think there are issues. However, Capitalism isn't markets.