Learn Italian: Non parlo con loro. - I do not speak with them.

Because it comes right after a preposition (con). After prepositions like con (with), per (for), or a (to), Italian uses "tonic" pronouns. The tonic pronoun for "them" is loro.

Does loro change depending on the gender of the people I am talking about?

No! Unlike some other pronouns, the tonic pronoun loro is exactly the same for masculine, feminine, and mixed groups.

Could I say Io non parlo con loro?

Yes, you can. However, Italians usually drop the subject pronoun (io) because the verb ending in -o (parlo) already makes it perfectly clear that "I" am the one speaking. Adding io here would add extra emphasis, as if saying "I don't speak with them (even if others do)."

Why is non placed at the very beginning of the sentence?

In Italian, the negative word non must always go directly before the verb it negates (parlo). Since the subject io (I) is left out, non ends up being the first word in the sentence.