Yes, it is possible to be conscientious and carefree. Conscientiousness at your job means that you do the best you can all the time to fulfil your duties.
But you can also be carefree, in that if you know that you did your best and fulfilled all your duties, you should not have to care what others think of you.
You are worrying too much about the situation with your manager, and you are trying to guess what that person is thinking.
All you can do is try to do your job to the best of your ability, and if you get passed over for promotion, or for the opportunity of a different job, just let it go and try not to dwell too much on blaming yourself.