Is the apartment applied for lawful or not, knowing that applying for it requires the applicant to be a resident renter, and an old rental contract was submitted? And is the son sinful for helping his mother with that?

If the state only provides low-cost apartments to those who do not own a home, and your mother owns a home, then it is not permissible for her to resort to deception to obtain an apartment, nor is it permissible for you to assist her in cheating. If she acquires it through such means, it will not be permissible for her. You must be dutiful to your mother, enjoin her to do good, and forbid her from evil gently and courteously, without harshness.