Answer 1

Oh wait. Wait. Wait. I got the problem.
Your numbers are not numbers (as in integers), they are strings!

Answer 2

When numbers are inputted the program works fine.
What you have handled wrongly is the no input case.

If (num1)||(num2==NaN) is what your 3rd condition does. You need to do:
if (num1==NaN||num2==NaN)

Answer 3

Jack Rehfeldt Look at my edited answer

Answer 4

I remember one neat "hacks" which is simply,
add a +

Edit:
var prompt1 = +prompt("Hey");
var prompt2 = +prompt("There");

Like that