It bothers me that Sun confided to Juliette that she was with another man besides Jin. Sun has not told another soul even Kate who she is close too. It doesn't make sense that she would confide in Juliette who is an Other.

I agree. Why give her anything she can use against her in the future? I feel certain that Juliet wouldn't hesitate to break the doctor-patient confidentiality pact if it served her purposes.

I think the point was to build our like/dislike of Juliette. We saw how earnestly concerned she was with Sun - and because of Sun's willingness to confide in her, the audience is meant to trust her again too.
Then we see her little message to Ben, and have mixed feelings.

Also, consider that Juliette, who is an other, still knows more than Kate does about the subject.

I think it was a woman DR thing - she needed to confess that she was afraid of when the DOC was - so she told Juliet why she was worried about having the sonogram.
She needed to confess - and Juliet knew all along so she set the stage to let Sun get it out -
I just wish Sun hadn't fallen for it!