You can't make people like you. You can be someone they like but too often I see employers on to strict or too nice. This brings me to bribing. Those carrots you throw out to "motivate" are bribes and in very bad taste. You aren't managing anything you are actually undermining your authority.
There is nothing wrong with team goals and rewards. In management, the goal is to have people trust you. When you remove yourself to: "who can sell the most" or "least injuries this month" you are saying you are pretty useless. You can't say who can sell the most I'll help you that would be cheating in the game you set up.
Aside from the fact some will naturally dig in their heels and not play you are not building a team. If you have a naturally competitive group then this type of motivtion may work for a bit. Most people are not so driven as to want to bragging rights.
The better way to handle this is a group goal for all to reach within a department. Attainable and encouraging people to come together and come to you with issues. This will go further in building a team and giving you chances to show you are a leader.