"1. Do I need to prepay state tax as well?"
I can only speak to California which pretty much mimics federal law in this area. I'd guess most or all states with an income tax would also require estimated tax payments.
"2. Is it enough if I assume 15% tax on capital gain and 30% on short term gain?"
I'd say "yes", for federal purposes, but understand that these issues always depend on your personal facts and circumstances.
"3. What if the sale happens in Q4 (or more extremely, nearly year end)? Since I would have no time to prepay, will I be charged penalty?"
The last estimated tax payment is due on 1/15 of the following year, so you'd certainly have time.
