I was a board member and a volunteer for our local rural H4H about ten years ago. What I saw might not be true in every affiliate, since they do run independently, but all are supposed to follow the basic guidelines set forth from the national office.
Yes, recipients of a house do get an interest free mortgage, but they have to meet basic income guidelines and be able to repay the principal. We did credit checks on potential families and several were rejected right away because of serious credit problems (unpaid student loans gone to collection, defaults on car loans, etc.) The family selection committee also looked at a family's monthly budget, and some applicants were also rejected based on those numbers--they would not be able to afford the payments in addition to their other bills.
As far as the "sweat equity" required, families entered into a covenant (I called it a contract once and was promptly scolded for calling it such a thing--I guess the national affiliate felt that a "covenant" was a more binding religious agreement than any contract would be, but potential families were not excluded based on religious beliefs.) The basic rule was that families put in some set number of hours on their own house, and then that same number on the next house, usually a hundred hours.
Also as far as I know, houses are required to meet state and local building codes and pass any house inspections required in the communities where the houses were located. Our chapter refurbished existing houses, either donated to the group or purchased very inexpensively. They needed extensive work to make them habitable, and during my tenure, the work was done by experienced volunteers, who were not licensed plumbers or electricians, to my knowledge.
There were also national guidelines as to square footage of houses, to reinforce the belief that recipients were to have modest and affordable housing. Townships did not offer any tax breaks, so school and property taxes were paid at the same rate as other homes assessed for the same amount.
One thing already mentioned that I would have really liked to see added to the H4H program was a home maintenance course and a home budgeting course, mandated as part of the agreement between organization and family.
That people living in H4H housing are suing the organization is nothing short of appalling, especially since the family is responsible for all maintenance on the property once they take ownership of it. What a slap in the face to the organization, and especially to volunteers who donated countless hours of their own time to a home renovation or building project.