Mi piace leggere la rivista settimanale al mattino.

Because in Italian piacere works “in reverse.” Instead of saying “I like X,” you literally say “X pleases me.”

• Mi is the indirect object pronoun meaning “to me.”
• Piace is the third-person singular form of piacere, agreeing with the thing you like (in this case, leggere).

After verbs of liking (“piacere,” “adorare,” “detestare,” etc.) you use the infinitive to talk about actions you like in general. So you say mi piace leggere (I like reading), mi piace nuotare (I like swimming), and so on.

La is the feminine singular definite article because rivista (magazine) is feminine.

• Feminine singular nouns take la.
• Settimanale is an adjective and agrees in gender and number (feminine singular) with rivista.

Yes, but the nuance changes:

• Una rivista settimanale = “a weekly magazine” (any one).
• La rivista settimanale = “the weekly magazine” (a specific one, or the concept of a weekly magazine in general).

All three can convey “in the morning,” but:

• Al mattino is very common to indicate a habitual action at that time of day (“in the mornings”).
• Di mattina also works (“in the morning”), perhaps a bit more general.
• Nella mattina is grammatically possible but less idiomatic; Italians prefer al mattino or di mattina for routines.

Absolutely. Ogni mattina means “every morning,” so you could say:
Mi piace leggere la rivista settimanale ogni mattina.