Después de cenar, devolvemos el libro a la biblioteca y sacamos otro.

In Spanish, after the preposition después de, you must use the infinitive: después de + infinitivo → después de cenar.

• Después cenar is wrong because the preposition is required.
• Después de cenando is wrong because gerunds don’t follow prepositions like this; the gerund would suggest simultaneity (e.g., cenando = “while eating”), not sequence.

Yes. Both are correct:

• Después de cenar focuses on the action (“after having dinner”) and is very natural for routines.
• Después de la cena treats dinner as a noun (“after dinner”). Use whichever fits your style; both are common in Spain.

Spanish normally drops subject pronouns because the verb ending already shows the subject. Devolvemos and sacamos end in -mos, so it’s clearly “we.” You can add it for emphasis or contrast:

• Nosotros devolvemos el libro… y sacamos otro. If the speakers are all female, nosotras is the form, but the verb still looks the same: devolvemos, sacamos.

The present here expresses a habitual or scheduled action (“we (usually) do this after dinner”). Spanish often uses the present for near-future plans too when a time reference is present.

• Future is possible: Después de cenar, devolveremos… but it sounds more formal or predictive. For routines, the present is natural.

Yes. In a library context, sacar (un libro) means “to check out/borrow.” In Spain, you’ll also hear:

• coger (un libro) prestado (fine in Spain; be cautious outside Spain where coger can be vulgar)
• pedir un libro prestado, sacar en préstamo
  All mean you borrow a book from the library.

You need the article. Spanish uses articles much more than English.

• Devolvemos el libro (a specific book already known in context).
• Devolvemos un libro (some book, non-specific).
  But bare nouns like devolvemos libro are not idiomatic here.

Yes. Use the direct object pronoun lo (for masculine singular):

• Después de cenar, lo devolvemos a la biblioteca. If you use an infinitive/gerund or an auxiliary, you can attach it:
• Vamos a devolverlo a la biblioteca.
• Estamos devolviéndolo a la biblioteca.

It agrees in gender and number with the implied noun:

• otro (masc. sing., e.g., libro)
• otra (fem. sing., e.g., revista)
• otros/otras for plurals.

Yes, devolver is o→ue in the present, but not in the nosotros/vosotros forms:

• yo devuelvo
• tú devuelves
• él/ella devuelve
• nosotros devolvemos (no change)
• vosotros devolvéis (no change)
• ellos devuelven
  So devolvemos is the only correct nosotros form.

Use the preterite:

• Después de cenar, devolvimos el libro a la biblioteca y sacamos otro.
  Note: sacamos (nosotros) is the same in present and preterite; context or adverbs like ayer, anoche clarify the time.

Yes, especially when the subject changes:

• Same subject: Después de cenar, devolvemos… (simplest)
• Different subject: Después de que mis hijos cenen, devolvemos…
  With después de que, use:
• Subjunctive for future/not-yet-realized: Después de que cenemos, devolveremos…
• Indicative for past/completed or factual: Después de que cenamos, devolvimos…
  In everyday Spain Spanish, for routines with the same subject, Después de cenar is most natural.

Yes, Spanish word order is flexible when clarity is preserved:

• Devolvemos el libro a la biblioteca después de cenar y sacamos otro.
• Después de cenar, a la biblioteca devolvemos el libro y sacamos otro. (marked/emphatic) Default order (Después de cenar, devolvemos el libro a la biblioteca y sacamos otro) is the most neutral.