Is money collected for performing the Hajj obligation through a "jam'iyyah" (a group in which each individual contributes an equal monthly amount, and one of them receives the full amount each month) considered a debt, and is the Hajj performed with it valid?

It is permissible to perform Hajj with money from a "Gam'iyyah" (cooperative savings fund) because the money from the Gam'iyyah is considered a loan. It is permissible for the one who takes the money to perform Hajj with it, as long as they are able to repay the loan when its due date arrives.