Is Jessie and James from Team Rocket boyfriend and girlfriend?

Answer 1

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.

Answer 2

Although they went to school together after James ran away from home and eventually joined a bike gang, Jessie and James parted ways afterwards. Jessie joined Team Rocket first, and James became her partner later on during their training days at Team Rocket Academy. They were probably in their mid to late teens at the time.

Answer 3

In the episode "Training Daze."