I've learned today that, in passato prosimmo, you don't conjugate verbs used with AVERE. For example, "Ho mangiato, Abbiamo mangiato... (qualcosa)". But if you want to use a direct object pronoun, the verb is conjugated according to the object (not the subject) of the sentence. For example, "Li ho preparati (I prepared them), L'ha fatta (She made it) (la lasagna, per esempio), L'ho mangiata (I ate it) (la torta, per esempio)". First of all, is this correct?

And my second question is, what about passato prossimo used with the verb AVERE? Same rule? Can you give me some examples? Lastly, does this rule only apply when you use a direct object pronoun, or also when you use an indirect object pronoun?

You must conjugate the verb according to the object only if there is a pronoun like <lo>, <la>, <li>, <le> (and <l'> if there's a vowel), like in the examples you mentioned. In all the other cases there's not a rule.

For example, you would easily find "Mi ha REGALATO dei fiori" and not "Mi ha REGALATI dei fiori", although the second one should be correct (maybe you can find it in literature, but it's VERY rare).
I can give you another example taken by Pirandello, a writer from the 20th century: "abbiamo SCELTE le più belle [piante]" = "We chose the most beautiful plants", where "le più belle [piante]" is the object and it is feminine. In daily life we never say this one, we use "SCELTO", it would be strange otherwise.

On the other hand, it is common to say "Mi ha ACCOMPAGNATA a casa", instead of "Mi ha ACCOMPAGNATO a casa" if the person who is talking is a woman ("mi", the object, is feminine), but you can use both without problem.

Indirect object is something different, you don't conjugate the verb according to it.