I would like to learn the Slovak language. Does the library have any lessons, on audio book or tapes? If not, can the librarian find these at another library that I could then access?

You could also get a library card and have access to their online resources, including Mango, which has many, many language resources on it

They might have something in the world languages section. But a librarian could definitely help you locate resources through another library in or out of state.

Yes, you can ask the librarian, and they will help you to the best of their ability.