I have limited information about this which I can share rest I hope some Gursikhs on the forum would be able to answer the questions.
Both Bhai Mati Das Ji & Bhai Sati Das ji were brothers and they were born in a Sikh Family. Their ancestor Bhai Gautam Dass became Sikhs since the time of Sri Guru Nanak Dev Ji (some historians say he was initiated into Sikhism by Guru Arjan Dev Ji) and since then the family had been devouted Sikhs. Bhai Gautam Das's son Bhai Praga was a commander of jatha in Guru Hargobind Ji army and had served all Guru Sahib since Sri Guru Angad Dev ji. Bhai Praga ji's grandsons Bhai Mati Das & Bhai Sati Das had served Guru Har Rai Ji, Guru Harkishan Ji & Guru Teg Bahadur Sahib Ji.
I have heard that they are from Chibber brahmin family but haven't done much research. I haven't read much about Bhai Dyala Ji but I have heard that he is from the same family of Bhai Mani Singh Ji (maybe his brother). So the full name can be found if you know full name of Bhai Mani Singh ji before getting Khande-ki-pahul.