Much, much simpler to just give everyone a simple cash payment (call it a tax credit, make tax payments due and refunds distributed monthly instead of annual, and you need no additional administration overhead), funded by a flat tax (of that credit amount over the mean income, so the math automatically balances out and it is revenue-neutral).
No. Flat taxes are inherently regressive because people with less money spend more of their income on taxes on necessities, and you are taxing them twice; once when they get paid, and again when they buy the things they need. Graduated tax scales are inherently fairer because those who derive the most benefit pay the most taxes.
If you want to fix taxes, start taking out exemptions. The best one is social security, which we call a contribution but is really just a kind of tax anyway. Remove the salary cap on social security and bang, you fix its funding problems immediately. Of course, that will drive people towards hiding more of their money in corporate tax dodges so that we have to crack down on those more, but that's not impossible either. Remember when the USA was just ignoring all the tax havens?