Do I report my 17 year old son's social security benefit he received for 6 months on my taxes?

No you do not enter a dependent's Social Security Benefits on your tax return.  If the dependent has no other income they do not report the benefits on a tax return.

How about if he made $4,700.00 from Door Dash?  Then do I need to report it?  Also, should he file his own tax return?  He wants to, he thinks he will start getting stimulus money himself since he will be in college more than 6 months of the year in 2021, and he is paying for college himself?

If your son made $4,700 form Door Dash he has to file his own tax return. This income considering self-employment and must be reported on Schedule C

Please follow the directions below to enter your Door Dash income:

1. Click the Federal tab on the left side of the screen.
2. Click Income & Expenses from the top of the screen.
3. On the next screen, scroll down to See all income.
4. Answer Yes to "Did you have any self-employment income or expenses"?
5. Continue through the interview to enter your business information.

Check the tax tips for delivery drivers:

Filing Tax Returns for Delivery Drivers: Tips and Advice