Much of these are already laws (or close to it.)
Your drivers license isn't enough, you also need a NICs check.
You do have to appear in person, though "original citizenship docs" isn't a requirement, naturalized citizens can legally buy guns.
If your name on the Form 4473 (or Form 1, Form 4, etc) isn't an exact match to your ID, you indeed cannot buy the gun. Same with your street address actually.
You can "apply" by mail if that means "buy online," but you still have to get it shipped to an FFL and do your NICs check and present ID in person.
If you've lost your birth cert/SSN card, you do have to get those replaced at your cost (iirc you may get a few free SSN card replacements). You don't need them to buy the gun but you do for carry permits in most states that still require permits to carry.
If the government does make a clerical mistake, it is up to you to correct it. Clerical errors here are rare though, it's usually you who made an error on the forms.
There is no deadline, nor election, but you will have to resubmit, which sucks for like a Form 1 at an 8mo wait right now, that's another 8mo to wait just because you filled the form out wrong. At least as of recently it isn't $200 to submit one, but still.