Is Jessie and James from Team Rocket boyfriend and girlfriend?

No, they aren't at a stage where they are ready to commit to another person. James is terrified at the thought of being in a relationship (to the point he ran and climbed up a tree in one episode) due to his being engaged against his will as a child, and the current object of Jessie's affections is herself. Meowth believes they "don't need the opposite sex because they have each other" and has informed them that they "act love-love all the time", but neither one of his two human companions seem to recognize their feelings for one another. They act like squabbling siblings through a large part of the series.