Is our solar system moving? If so, is our solar system being attracted closer to any other solar systems?
Yes, but in a relative sort of way.
Our solar system is moving along with the galaxy of course. It is also moving, though rather slowing, within the galaxy. We do not have a good grasp of other existing planetary systems to be able to answer the second part of your question. But since the nearest star is many light years away, I would not concern myself with that too much.
Yes, and yes,
All the stars move around the center of the galaxy, as planets move around the Sun, This is the main movement and. However, every star as a component of the velocity which affects the distance between stars. So there are stars which are aproaching, and others which are distancing. But the distances between stars is so huge that is unprobable that the stars become close enough to be dangerous. The star Gliese 710 will pass about 1 light year from the Sun into 1.5 million years.