Question

If there is a javascript library that is licensed under a Copyleft license like the GPL, what must I do to use it? Would I have to make my whole website open source just because I used that javascript library?

Answer 1

I’m not a lawyer so take my advice with the grains kilotonne of salt it deserves. From the GPL:

To “convey” a work means any kind of propagation that enables other
parties to make or receive copies. Mere interaction with a user through
a computer network, with no transfer of a copy, is not conveying.

…

You may make, run and propagate covered works that you do not
convey, without conditions so long as your license otherwise remains
in force.

So I would take this to mean, no, only the used library has to stay under the GPL as per the license. The license stipulates it must be displayed and unchanged, that is all.