Ho già spedito la cassetta con i libri antichi al restauratore.

In Italian, to talk about a completed action in the recent past you normally use the passato prossimo, which combines an auxiliary verb (avere or essere) with a past participle. Here:

• ho (from avere) + spedito (past participle of spedire)
  This corresponds to English "I have sent" or simply "I sent." Saying spedisco would mean "I send" (habitually or right now), not "I have already sent."

The most common position for già (meaning "already") is between the auxiliary and the past participle:

• Ho già spedito…
  You can also put it at the very beginning for emphasis:
• Già ho spedito… (means “Already, I have sent…,” more emphatic)
  Placing it after the participle (“Ho spedito già…”) is possible but less idiomatic.

Here con means "containing" or "with," indicating the contents of the crate:

• la cassetta con i libri antichi = “the crate containing the old books.”
  Using di (as in cassetta di libri) would more often describe material or origin (“crate made of books,” or “crate belonging to books”) and sounds unnatural for contents.

Al = a (to) + il (the). Italian uses the preposition a to mark the indirect object ("to the restorer"). You need the article il because restauratore is masculine singular:

• a + il restauratore → al restauratore.

La cassetta is a feminine singular direct object. The correct direct-object pronoun is la. In compound tenses the pronoun goes before the auxiliary, and the past participle agrees in gender/number with that pronoun:

• L’ho già spedita al restauratore.
  (Note: spedita ends with –a to match la.)

Both verbs mean “to send.” Generally:

• spedire is common for physical items (packages, letters, crates).
• inviare works for both items and messages (emails, invitations) and sounds slightly more formal.
  Yes, you can say Ho già inviato la cassetta… with the same meaning.

Yes. Without già you simply state the fact:

• Ho spedito la cassetta… (“I sent the crate…”)
  With già, you emphasize that the action has already been completed, perhaps earlier than the listener expected.

Absolutely. Italian allows flexibility:

• La cassetta con i libri antichi l’ho già spedita al restauratore.
  Here you front the object (la cassetta…) and use l’ as its pronoun before the auxiliary. You can also start with Al restauratore or Già, depending on what you want to highlight.