when did jake and dirk first start going out? and why did they break up exactly?

They started going out right after they entered.

And they broke up for a number of reasons. Dirk was possessive and overbearing which caused Jake to be distant and to avoid him. The nail in the coffin was during the trickster debacle Jake told Dirk that he was going to marry Jane and have like 50,000 babies and Dirk broke up with him.