Q. Because library eresources are not accessible to everyone browsing the Internet, is there more freedom for me to reproduce content from them, display content publicly, or otherwise use content in ways that might fall outside of copyright or fair use?

No. Paywalled items or material only accessible as an authorized user does not mean more leeway or broadened usage than you might otherwise expect. Because you buy a print book, that does not give you more right to do what you wish with the contents inside. That is not how copyright works. Treat licensed resources like you would anything you encounter online. Pay attention to licenses that tell you how material may be used as well as terms and conditions, and keep fair use in mind.