Can I make a queries to the transit part of the google maps API that
For a given GPS coordinate, give me nearby stations?
For a given stations, give me the next departures?
Google maps itself provides that information (see screenshot). But is this information accessible through the API? 
Not everything that the Google Maps app can do are readily available in the Google Maps Javascript API / Web services. So don't expect to be able to copy the Google Maps app features.
You can easily get transit_stations by using a Places Nearby Search.
See docs: https://developers.google.com/maps/documentation/javascript/places#place_search_requests
But you can only get departure times of a transit stop in the context of a Directions query. https://developers.google.com/maps/documentation/javascript/directions#TransitInformation
