Grandfather pays his grandson $200.00 a week using the term caretaker.
Grandson acts as self-employed, pays his taxes. But eventually most of the funds he gets, returns back into the household account. Is this strategy legal?

As long as grandfather files a 1099

Yes, it is legal. What the grandson does with the money is strictly up to him. If he wants to give it back to his grandfather, he can do so.

As long as the taxes are paid and the Form 1099-MISC is filed, they are good to go!

I imaging the grandfather deducts the social security and medicare from
The grandson cash?

"Deducts social security and Medicare from The grandson cash?" For whom? You said the grandson pays his taxes.

Oh so it's the grandson that set aside deductions. Not the payee the grandfather.
That is okay too.

Actually, the grandfather wtihholds Social Security and Medicare, then matches it dollar-for-dollar, then sends it in with the Form 941 on a quarterly basis

Oh I see. The deductions for taxes and FICA are about $47.00. So the grandfather would match this amount and sent the total amount to the authorities

That is correct!