I heard this question mentioned during last weeks Broncos/Texans game. It was in reference to Mike Shanahan and the fact that he was one of three coaches who had coached a team 10 years and had more championships than losing seasons. (link) Too bad for Shanahan that his time in that illustrious group is about to come to an end. You see the Broncos are 6-9 and with only one game left in the season and no hope of making the playoffs, Denver is already a lock to have a losing record this year. Good thing I asked this question now and not in two weeks!
The second head coach that has coached a team 10 years and had more championships than losing seasons is the immortal John Madden. Madden won one Super Bowl as the coach of the Oakland Raiders and never had a losing season while their coach. It’s a wonder he didn't coach them for longer. Did you know this Hall of Fame coach's .759 winning percentage during the regular season ranks as highest ever among coaches with 100 career victories? (link) The guy was a coaching legend. It's a shame people my age only know him for his video games and his time in the broadcast booth, even though he very well could have gotten into the Hall of Fame for either of those contributions if he had never coached a game.
The third coach that no one was able to come up with was Hall of Fame member and coaching legend Paul Brown. While the Cincinnati Bengals’ stadium bares his name, Brown was elected into the Pro Football Hall of Fame before he ever coached for Cinci. Brown forged his coaching legacy while leading the Cleveland Browns to a 167-53-8 record, four AAFC titles and three NFL crowns in 17 years with only one losing season. (link)
The bonus question's answer is another Hall of Famer, Joe Gibbs. There was some confusion here, but the question said the head coach had to have coached a team 10 years, it did not say later non-consecutive seasons were not counted. The first time Gibbs coached the Washington Redskins he had 10 winning seasons, 1 losing season, one 8-8 season and 3 Super Bowl victories. But, Gibbs couldn’t stay away from the game he loved and came back to coach the Redskins in 2004. Since that time he's had one winning season and two losing seasons bringing his overall time with the Redskins to three championships and three losing seasons. Gibbs shouldn't fret too much because with one more victory this weekend he will be leading his 'skins back to the playoffs.
One coach that spent over 10 seasons with one team and definitely had more championships than losing seasons with that team is Curly Lambeau, founder of the Packers and its first head coach and playing captain. He won 6 championships while coaching the Packers and had only 1 losing season. I know Mark mentioned him in his answer, but everywhere I've looked they only list Brown, Shanahan and Madden in the answer. The only reason I can think of for his exclusion is that while he coached over 10 seasons with the packers and won 6 championships, he also had 6 losing seasons throughout his time as a head coach in the NFL. The question doesn’t say the losing seasons all have to be with the same team. I know this is a technicality and I can't find it explained anywhere, but this is the only rationale I can come up with for his exclusion from the answer.
Thanks to everyone who submitted an answer, and I'll try to come up with a non-sports related question for next time so others can participate!