Allocation of mortgage interest deduction between joint homeowners

It does actually specify how shares are determined. Look at the quote again:

Show how how much of the interest each of you paid, and give the name and address of the person who received the form. Deduct your share of the interest on Schedule A (Form 1040), line 8b, and print “See attached” next to the line.

The shares are determined based on the actual payments. How much each of you paid. So if you split payments equally then you also split the mortgage interest deduction equally. The legal authority is the IRC Sec. 163:

There shall be allowed as a deduction all interest paid or accrued within the taxable year on indebtedness.

(Bonus question: is there a legal way for A to deduct the entire $30,000 despite only actually paying $20,000 out of pocket? For instance, suppose that B gifts $20,000 to A (ignoring any gift tax implications), and A then pays the entire mortgage for the year and claims the full deduction.)

I would take that to a lawyer. Gifts are only gifts if there are no strings attached, here there are strings attached and as such it could be argued by the IRS that it wasn't a gift but rather a payment by proxy.