Is it permissible to take money from a government company designated for projects, to be paid back in installments, knowing that the purpose is not a project but rather to complete the building of a house?

The reference for taking a loan is the condition set by the lending party. If it stipulates that the loan is for a commercial project, then it is not permissible to take it for anything else, such as building a house, because the Prophet, peace and blessings be upon him, said: "Muslims are bound by their conditions, except a condition that forbids what is lawful or permits what is unlawful." Conditions in contracts are valid and binding as long as they do not contravene the Sharia.