I don't know the legal framework for RSUs, so I'm not sure what is mandatory and what is chosen by the company issuing them. I recently reviewed one companies offering and it basically looked like a flat purchase of stock on the VEST date. So even if I got a zillion shares for $1 GRANTED to me, if it was 100 shares that vested at $100 on the 1st, then I would owe tax on the market value on the day of vest. Further, the company would withhold 25% of the VEST for federal taxes and 10% for state taxes, if I lived in a state with income tax. The withholding rate was flat, regardless of what my actual tax rate was.
Capital gains on the change from the market value on the VEST date was calculated as short-term or long-term based on the time since the VEST date. So if my 100 shares went up to $120, I would pay the $20 difference as short term or long term based on how long I had owned them since the VEST.
That said, I don't know if this is universal. Your HR folks should be able to help answer at least some of these questions, though I know their favorite response when they don't know is that you should consult a tax professional. Good luck.