I found this information with the Lodz Ghetto Inhabitants database; from there I found the Lodz Ghetto List, which had Mojsze Kuczynski listed on the page displaying 751-800. His birthday (1927) fits the period of School #5 (1926 - 1934).
I found the possible family members by sifting though the Lodz Ghetto List found from the Lodz Ghetto Inhabitants database and searching for people with the same surname and the same address. The dates of the possible parents are plausible (1903 for Rywka and 1899 for Bejrysz). As for Mojsze's siblings, I am not possitive, as there are only ten months between their births. Although it is possible, it is also a bit suspect due to the closeness of the births; Szmul was born August 11th 1924 and Lemel was born June 06th 1925.