Learn Hardcore German: Die Mitgliedschaft in der Bibliothek ist kostenlos, aber eine Anmeldung ist notwendig. - The membership in the library is free of charge, but a registration is necessary.

Because Mitgliedschaft is a feminine noun in German. All German nouns have a grammatical gender (masculine, feminine, or neuter), and Mitgliedschaft requires the feminine definite article die.

Why do we say in der Bibliothek instead of in die Bibliothek?

The preposition in can govern either the dative or the accusative case.

Use the dative (here der Bibliothek) when you talk about location (where something is).
Use the accusative (i.e. in die Bibliothek) when you talk about movement toward a place (where something is going).
Since you’re stating where membership exists (location), you need the dative: in der Bibliothek.

Why is kostenlos not declined (e.g. kostenlose)?

Because kostenlos is used predicatively after the verb sein (“ist kostenlos”), not attributively before a noun. Predicative adjectives in German always appear in their base (undecorated) form:

Predicative: Die Mitgliedschaft ist kostenlos.
Attributive (before a noun) you would decline: eine kostenlose Mitgliedschaft.

Why does the sentence use eine Anmeldung with an article instead of just Anmeldung?

Here Anmeldung is a countable noun meaning “a registration” or “one registration.” You need an indefinite article (eine) to indicate one instance of that noun. Omitting the article (or using the plural Anmeldungen) would change the nuance:

eine Anmeldung = one registration is necessary.
Anmeldungen (plural) = registrations in general, which doesn’t fit this context.

Why is Anmeldung capitalized?

In German, all nouns are capitalized, regardless of where they appear in a sentence. Anmeldung (registration) is a noun, so it must be capitalized.

How does the coordinating conjunction aber affect word order in the second clause?

aber is a coordinating conjunction, so it does not send the verb to the end (unlike subordinating conjunctions such as weil). It simply connects two main clauses, and each clause keeps the normal German word order (verb-second). Thus:

first clause: Die Mitgliedschaft (subject) – ist (verb) – kostenlos.
second clause: aber (conjunction) – eine Anmeldung (subject) – ist (verb) – notwendig.

Could you use Registrierung instead of Anmeldung, and is there any difference?

Yes, Registrierung (from the verb registrieren) is also correct and understood. However:

Anmeldung is more common in everyday German, especially for library or event sign-ups.
Registrierung sounds a bit more formal or technical, often used in IT contexts (e.g. website registration).