Learn Russian: Я читаю и никому не мешаю. - I am reading and not disturbing anyone.

The verb мешать (to disturb, to bother) requires its object to be in the Dative case. Think of it as meaning "to be a hindrance TO someone." Никому is the Dative form of никто (no one).

Why are both никому and не used here?

Russian uses mandatory double negation. Whenever you use a negative pronoun like никто or its other case forms, you must also put не (not) in front of the verb.

Is it okay to change the word order and say Я читаю и не мешаю никому?

Yes, that is grammatically correct. However, putting the negative pronoun right before the не + verb (like никому не мешаю) is the standard, most natural word order in Russian.

Why don't we need to repeat я (I) before мешаю?

Just like in English, if the same subject is doing both actions connected by и (and), you don't need to repeat it. The -ю ending on мешаю also makes it completely clear that "I" am the one doing the action.