Learn Polish: Posłuchajże mnie, to mnie bardzo irytuje! - Do listen to me, this irritates me very much!

It adds a strong sense of urgency, impatience, or emphasis. While posłuchaj is a standard command meaning listen, adding the emphatic particle -że turns it into do listen or just listen to me. This perfectly matches the speaker's irritation expressed in the second half of the sentence.

Is there a rule for when to use -że versus its shorter form -ż?

Yes! You use -że when the imperative verb ends in a consonant, such as the -j in posłuchaj. If the imperative ends in a vowel, you use the shorter form -ż, for example in the plural form posłuchajcież.

The word mnie appears twice here. Is it serving the exact same grammatical function both times?

No, they are actually in two different cases! The verb posłuchać (to listen to) requires its object to be in the Genitive case, so the first mnie is Genitive. However, the verb irytować (to irritate) takes a direct object in the Accusative case, making the second mnie Accusative. Fortunately for learners, the pronoun ja uses the identical form mnie for both.

Why does the speaker use the perfective posłuchajże instead of the imperfective słuchajże?