Learn French: Le bibliothécaire est très sévère et donne un avertissement. - The librarian is very strict and gives a warning.

When describing people, sévère usually means "strict" or "stern" rather than the English "severe" (which we typically use for things like storms, injuries, or extreme punishments).

I always confuse library and bookstore in French! Is bibliothécaire related to one of those?

Yes, it's related to bibliothèque (library), making a bibliothécaire a librarian. The classic false friend to watch out for is librairie, which actually means "bookstore" (run by a libraire, or bookseller).

Does bibliothécaire change spelling if the librarian is a woman?

No, because the noun already ends in an -e, the spelling stays exactly the same. You only change the article: le bibliothécaire for a male librarian, and la bibliothécaire for a female librarian.

What verb is the noun avertissement (warning) based on?

It comes from the regular -ir verb avertir (to warn). Adding -issement to the root of an -ir verb is a common way to turn an action into a noun in French.

Since this lesson is about sinon (otherwise), how could I use it to continue the thought in this sentence?