Learn Russian: Они меня не слышат.

меня is the Accusative case form of the pronoun я (I). In Russian, the direct object—the person receiving the action—must be in the Accusative case. Since "they" are doing the hearing and "me" is the person being heard, we use меня.

Why is the word order Они меня не слышат instead of Они не слышат меня?

Both are perfectly correct! Russian word order is flexible. It is very common to put short object pronouns like меня right before the verb, but if you say Они не слышат меня, a native speaker will understand you just as well.

The dictionary form is слышать, so why does the "they" form end in -ат instead of -ют?

слышать (to hear) is an exception to the normal verb rules. Normally, verbs ending in -ать use endings like -ют for the Они (they) form. However, слышать follows the pattern of verbs ending in -ить. That is why it takes the -ат ending here.

Does the negative word не always go right before the verb?

Yes, when you want to say that an action does not happen, не must immediately precede the verb it negates. You cannot put меня between не and слышат.