[Giants]

1994 Orioles hold the record for fewest errors by a major league team with 57.  

Giants lead the majors with just two errors after 12 games, which projects to 27 errors over the course of the season.  Statistically unlikely.  And they haven't had an errors in their last eight games.  I couldn't find the record for most games without an error by a team, but right now the Giants have eight straight error free games.

SF had 11 straight errorless games in 2010.

Source:  Baseball Reference

As for the major league record, I could not find a definitive source for that information.

[Giants]

1994 Orioles hold the record for fewest errors by a major league team with 57.  

Giants lead the majors with just two errors after 12 games, which projects to 27 errors over the course of the season.  Statistically unlikely.  And they haven't had an errors in their last eight games.  I couldn't find the record for most games without an error by a team, but right now the Giants have eight straight error free games.

SF had 11 straight errorless games in 2010.

Source:  Baseball Reference

As for the major league record, I could not find a definitive source for that information.