JIK, most libraries I know of only stop you from going to places that are subscription-based, such as Ebsco or Proquest. And if you are registered with that library, you can access there as well. This will continue, I think, until the pay basis for these sites (population, # of registered patrons, or whatever they use) changes.

Nowhere do I know of a library that doesn't let you check its online catalog.

Because I know that search engines differ, sometimes significantly, I use several. I think that they offset each other's faults.

Usually what I'm looking for is quick information, not in-depth research. Doing REAL research requires a whole new kit of tools, and includes print materials.