Learn Italian: L'hai già scaricata, l'ultima puntata? - Have you already downloaded it, the last episode?

This is called right dislocation, a very common feature in spoken Italian. You use a direct object pronoun like L' early in the sentence to smoothly connect to the context, and then tag the actual noun (l'ultima puntata) at the end to clarify or emphasize what that pronoun refers to.

What exactly does the "L'" stand for here? Is it "lo" or "la"?

It stands for la, because it refers to the feminine singular noun l'ultima puntata. When the direct object pronouns lo or la come right before a form of avere that starts with an h or a vowel, they drop their own vowel and become l'.

Why does the past participle "scaricata" end in "-a" instead of "-o"?

Normally, verbs conjugated with avere in the passato prossimo end in -o. However, when a direct object pronoun (lo, la, li, le) comes before the verb, the past participle must agree with that pronoun in gender and number. Since L' here stands for the feminine singular la, the participle becomes scaricata.

Is there a specific reason "già" is placed in the middle of the verb?