I'm guessing you don't watch Dr. Phil, right? He's done a whole slew of shows on this type of topic and makes the point that this situation benefits you (the parent) to the detriment of the child. Dr. Phil's position is that you are feeding your guilt to make yourself feel better, but this hurts your child. This is just a hunch, but how much does your 20 year old help with watching your youngest? And, if you did not have a seven year old in the home who benefits from the extra supervision, would have already given the 20 year old his walking papers?
He will not be happier being independent but being independent is an important step in growing up. Your job is to help him grow into a man . . . and a MAN does not live at home with his mommy while refusing to pay rent and/or do chores.
My husband handled this a couple of years ago with our 18 year old charge by offering to drive him to the (Army) recruiter's office. His philosophy was that if you want someone to take care of you (food, clothing, housing, medical) then you're not going to do it without giving something back. And a drill instructor is NOT going to be sympathetic to complaints about chores!
Your son is walking all over you and obviously does not respect you, otherwise he would not be refusing to pay rent because you expect him to do chores. Whatever you decide to do, I would make certain you are emotionally able to follow through with it because if you don't follow through with the consequence you will lose even more credibility and respect.
I personally would give him a written three day notice to pay rent or vacate. (This would include the exact amount he has to pay in past-due rent and it gives him the option of paying it within those three days or moving out.) I would also raise his rent to the amount that he would have to pay to rent a room in any other private home and would make it crystal clear that he WILL do reasonable chores around the house. I would put this into a written lease and would require him to sign it. I would also include a clause that this is a six month lease and that he will be expected to move out at the end of that lease unless he is enrolled in college AND making passing grades. I would absolutely evict him (while making it clear that he was invited to dinner at least once a week) if he failed to comply.
But if your son won't respect these rules and you have any concerns about enforcing the consequences, give him the required 20 day notice to move out and give him the walking papers on his life. (And don't give him any financial support to move out or get started in an apartment. If he doesn't have the savings, he can sell some of his many toys that he's been able to buy because he's been mooching off of mom for the last two years!)
On a personal note: I have a 30-something year old sister who bounces back and forth between my mother and father's basement. She has always worked full-time. She is smart but she is somewhat lazy and financially irresponsible. She moved out a couple of times, but she didn't like that her standard of living decreased when she had to support herself. She much preferred living in a nice home and having her entire paycheck to buy what she wanted. My parents did NOT do her any favors. She is a dependent child masquerading as an adult. She has no sense of self-worth. I think the thought of living alone terrifies her. I worry about who she will be when she's in her 50s - particularly since my parents will probably not still be alive (because they both have significant health problems now.)