Checks are issued only to those who have valid social security numbers. You can’t game the system by co-habitating.
Yes you can. easily.
A family with a “duplex” declared house suddenly gets 10000 per year vs 5000. (just example numbers the real numbers will be higher based on history and gradual inflation)
The neat part is that it will all be legal, all out in the open, and local municipalities will have every incentive to allow such “residence splitting” because it means money for them.