A polyfill is a short snippet to implement a missing feature. Notably almost all recent additions to JS and DOM have been designed to be polyfillable where possible. Generally that means under a thousand lines, by the way.
A library in this context means a set of functionality built on top of the platform, like jQuery, with its own API, etc. "Use a library" and "use a polyfill" are not the same advice.
Yes, it is.
> It's a dozen lines, not a thousand
A polyfill can be any size, but there's no size minimum to be a library.