It is "integral from a to b of f(x) with respect to x". Or you can say "It's the integral of f with respect to x and goes from a to b". dx means with respect to.
Not quite. You would say "the limit as n approaches infinity of the sum from i is equal to 1 to n of F of x i star times delta x". That's pretty nasty to just say it straight away.
You can make it easier for yourself by saying something like "the limit of the sum as n approaches infinity. The sum is from i equal to 1 to n of F of x i star times delta x "
It should totally be obvious, yes. In fact some theorems in analysis that deal with integrals of one variable sometimes don't even write the dx. However, it's good practice. Not to mention that it's difficult for calculators to "know" which variable you are integrating with respect to when you have arbitrary constants lurking in your expression.
However, if you look at this tool here integral 1/sqrt(1+x^2) - Wolfram|Alpha, I have inputted an integrand without the dx and the program recognizes it.
However, when you put in two variables like in this example integral 1/sqrt(x^2+y^2) - Wolfram|Alpha, it will just pick one of the variables and integrate with respect to it.