Leiden University and its student unions did arrange a multitude of social activities, none of which I ever attended. I did play tennis and go to the gym at the University Sports Center, which have a lot of other sports clubs to join.
You can pretty much do anything. There's a list on the USCLeiden website for sports and the Leiden University website has descriptions of all the clubs you can join.
Go to Belgium. I took a train ride down to Brussels and spent a weekend exploring the city with a friend. It's a beautiful and quirky town that have quite a lot of nice things to see. They also have waffles. Traveling to Paris is an option to continue on to, which I doubt I need to vouch for you to consider. I didn't go there however, I was busy in Brussels.