This boy recently lost his father. This deceased father never provided direction for life. This boy's mom has now replaced his departed father with another man. This new man is a pain in the ass, difficult, and unnecessarily harsh. He just does not understand.
This kid needs direction, a friend and a guidance counselor. He needs a father. At some point, the mom's maternal love for her children will overcome her need for the OP to support her. Then life will really become difficult but not if the OP becomes a father figure for her children.
Sit down, talk with him about options whether they are work, trade school, military, or community college. Explain that he needs to pick one and act on it. If action is not taken, life will become difficult in the home. Give him rules and responsibilities in the home. Give him an alarm clock and wake him up at 6am with cold water dumped on him in his bed if he does not act on any of the discussed options. If he is not in school, then he needs to be working. Disconnect the internet at night or remove his access from the internet.
Give him positive motivation by reminding him of the fun things in life that he cannot partake in with no money. How will he purchase his next car, his next computer, or get a girlfriend with no job and no money?
Forbid his loser friends from entrance in the home and explain why these friends are losers and not welcome. Get him to understand the traits that make these people intolerable.
Forcing him out of the house is not a solution. He is now the OP's responsibility, not society's. He will only get into worse trouble and more people will see the problems in this home as these problems become public. Police visitations will become uncomfortable with an illegal basement unit.