The Watchtower doesn't have a language library as such , though they do have lots of translators (5,200 at the last count) , most of whom ae involved in translating WTS publications from English into another language. They don't have any translators capable of translating original Bible Greek or Hebrew into modern languages though - they left that up to Fred Franz and other experts in the past.
Do the witnesses still believe that there will only be one language spoken in the new sysytem? One universal language?
Yes
What other institution would have such a collection of languages?
See above.
Do you think watchtower has shared their language information with other repositories?
No. The WTS doesn't have "language information"or any resource as such , they have teams of volunteer translators ( of various abilities) who get given a Watchtower or Awake , or new publication and told to translate it. A couple of translators post on JWD. The WTS wouldn't share any information as they don't have any to share , though I would imagine that they would use repositories and libraries for research purposes , as they did , controversially , in the case of the UN library.