Your final sentence and question is one of the most common questions asked in physics classrooms! The answer is that you do have weight and gravity is pulling the ISS towards the Earth. The ISS is in free-fall towards the Earth, and that is why it feels as if you are weightless inside the ISS. When you are in free-fall such as falling off a cliff, it feels as if you are weightless (this is of course an observable phenomenon). The reason the ISS never falls into the Earth is because it is going so fast perpendicular to the planet (the speed necessary is dictated by vOrbit in the above equations).

Here is a short video explaining the subject: