I want to know more about Yami. Is it believed that there was a twin sister of Yama named Yami?
The story is described in chapter 77 of the Markandeya Purana:
The daughter of Vishwakarman [the divine architect] was the wife of the sun, by name Sajna who begat on her a son, by name Manu, who, illustrious as he was, was equally a master of various branches of learning. He was the son of Vivaswat and was therefore called Vaivaswata. As soon as she was espied by the sun Sajna used to close her eyes and therefore he, in anger, addressed to her cruel words: "Since, O stupid one, you close yours eyes as soon as I cast my looks on you therefore will you give birth to Yama, the destroyer of creatures." Thereupon the goddess, stricken with fear, assumed trembling looks, at which the sun again said to her. "Since beholding me, you have assumed trembling looks you will give birth, as your daughter, to the fickle river." Thus by the imprecation of her husband she gave birth to Yama and the great river celebrated under the name of Yamuna. Henceforth she, with great difficulty, began to suffer the effulgence of the sun.
By the way, in case you're interested that isn't how the story ends: after that Sanjana leaves Surya, because she cannot take his brightness and she's replaced with her shadow Chaya, who pretends to be Sanjana. You can find out about the rest of the story in my answer here.