Yes it can—but it won’t do it much good. The reason is that Google’s servers aren’t fit for solving the Bitcoin mining problem in the same way that ASICs are. For reference, if Google harnesses all of its servers for the sole purpose of mining Bitcoin (and abandons all other business operations), it will account for a very small percent (less than 0.001%) of the total mining power the Bitcoin network currently has.
That’s why mining pools came into existence. The idea is simple: miners group together to form a “pool” (i.e., combine their mining power to compete more effectively). Once the pool manages to win the competition, the reward is spread out between the pool members depending on how much mining power each of them contributed. This way, even small miners can join the mining game and have a chance of earning Bitcoin (though they get only a part of the reward).
Because it's similar to gold mining in that the bitcoins exist in the protocol's design (just as the gold exists underground), but they haven't been brought out into the light yet (just as the gold hasn't yet been dug up). The bitcoin protocol stipulates that 21 million bitcoins will exist at some point. What "miners" do is bring them out into the light, a few at a time.
The domain name "bitcoin.org" was registered on 18 August 2008. In November 2008, a link to a paper authored by Satoshi Nakamoto titled Bitcoin: A Peer-to-Peer Electronic Cash System was posted to a cryptography mailing list. Nakamoto implemented the bitcoin software as open source code and released it in January 2009. Nakamoto's identity remains unknown.