¡Hola VR143!
(I've split this out into a separate thread so that its easier for other users to find.)
Thanks for your question! I've answered it below, and have added a section about the first sentence as well.
1. Using Para mí or Me
Si pudieras, ¿echarías estas cartas al correo para mí?
If (you) could, would (you) mail these letters for me?
Si pudieras, ¿me harías un favor?
If (you) could, would (you) do me a favor?
Whether you would like to use para mí or the indirect object pronoun me when you want to say “for me” is entirely up to you - either one is fine. You're just more likely to see and hear me used more often in common expressions like hacerme un favor “to do me a favor” simply because it's easier and faster to say.
2. Para mí vs. Por mí
Depending on your exact desired meaning, it might actually be better to change the first sentence to por mí rather than para mí:
Si pudieras, ¿echarías estas cartas al correo por mí?
If (you) could, would (you) mail these letters for me?
This is because por mí is used when you're asking someone to do something on your behalf. You do have the option of using para mí here, but you'd be implying that this would be done to to please you (kind of like it's a gift for you, in a way).
(Note that you cannot replace por mí by me - only para mí.)
I hope that this is helpful, and that it hasn't made things more confusing! Do let me know if you still have any questions.
If you'd like to see our explanations on replacing para + [pronoun] with indirect object pronouns, you can find that in Lesson 5.7 “'To Me' and ‘For Her’: Indirect Object Pronouns” under the section “About Prepositions.”
For more explanations on por vs. para, you can check out Lesson 11.6 “POR vs. PARA” and Lesson 15.6 “POR and PARA Revisited.”
Saludos,
Liss