Offering to pay the tax does trigger more tax; but as the added tax is less than what you add to the price, the total amount converges (as long as the tax rate is less than 100%). To have the recipient end up with a net payment N after taxes you've to pay N/(1-r), with r being the tax rate.

With a 25% tax rate and car value of $28,500; to make it a tax-free gift Oprah should've paid an additional $9,500.

Yeah, the term is "grossing up" and it's often done for bonuses and things. I've also seen it on sweepstakes. One for Fage yogurt offered a dream vacation to Greece and included a gross-up to pay the income tax.

Yes.

There’s a great YouTube view about exactly this by Stand Up Maths if you want to go down a rabbit hole.

https://youtube.com/watch?v=3cNdM7W0VlQ