Learn Italian: Marco mi fa perdere la pazienza. - Marco makes me lose my patience.

This is the causative construction, used when a subject causes someone else to do an action. You conjugate fare for the person causing the action (Marco fa) and follow it immediately with the infinitive of the action being caused (perdere). These two verbs act as a single grammatical unit.

Is the pronoun "mi" a direct or indirect object here?

It is an indirect object, which is a tricky rule of the causative construction! Because the second verb (perdere) already has a direct object (la pazienza), the person being forced to do the action must become the indirect object. So, mi here grammatically acts as "to me" even though we translate it as "makes me".

Could I attach the pronoun to the infinitive and say "Marco fa perdermi la pazienza"?

No, you cannot. Because fare and the infinitive act as an inseparable block in a causative construction, object pronouns must go before the conjugated verb (fare). Therefore, it has to be mi fa perdere.

Why does the Italian say "la pazienza" (the patience) instead of "la mia pazienza" (my patience)?

In Italian, when the "owner" of a state of mind, body part, or personal attribute is already clear from an object pronoun (like mi), you drop the possessive adjective. The definite article la combined with the pronoun is enough to convey "my patience."