Learn French: J'ai réparé la voiture moi-même.

It adds emphasis, exactly like "myself" in English. While J'ai réparé la voiture just states a fact, adding moi-même highlights that you did it personally—perhaps instead of paying a mechanic or asking a friend for help.

Can I place moi-même right after the subject, like "Je moi-même ai réparé..."?

No, you can't separate the subject pronoun (je) and the verb like that in French. Emphatic pronouns usually go at the end of the clause. If you want to add heavy contrast right away, you can put it at the very beginning, separated by a comma: Moi-même, j'ai réparé la voiture.

How would this sentence change if I wanted to say "I even repaired the car"?

To use même as the adverb "even", you drop the pronoun and the hyphen, and place it right after the conjugated verb. It would become: J'ai même réparé la voiture.

Since la voiture is feminine, why doesn't réparé end with an extra 'e'?

Because the direct object (la voiture) comes after the verb. When using the avoir auxiliary in the passé composé, the past participle only agrees in gender and number if the direct object is placed before the verb (for example, if you used a pronoun: Je l'ai réparée).