Day and night quilt.
 

How many blocks do I need for a Queen size Quilt?

this free pattern says it makes 12" blocks. http://www.quilterscache.com/N/NightandDayBlock.html
the suggested queen size quilt is 90x95. i would probably use 8 blocks by 8 blocks for 96x96
HTH

Eleanore Burns book says forty two blocks with a setting of 6 wide by 7 long for a queen size.

How do I calculate the border?

I would just use whatever size I would need to make the size you want. I have a tall mattress, I make mine long and wide enough to go a bit past that.

If you make it 96x96, you would have 2 strips cut 96 1/2" and 2 strips 96 1/2 + twice the width of your border.