Learn Portuguese (Portugal): Eu mesmo imitei a voz daquele cantor.

By placing mesmo directly after the pronoun eu, it limits its scope to the subject, acting as an intensifier that means "myself". It emphasizes that the speaker personally did the action, rather than someone else.

Would mesmo change if the speaker were a woman?

Yes! When modifying a noun or pronoun to mean "myself" or "herself", it must agree in gender and number. A female speaker would say Eu mesma imitei. If it were placed later in the sentence as an adverb meaning "really" or "even", it would stay as mesmo regardless of the speaker's gender.

How would the meaning shift if we moved it to say Eu imitei mesmo a voz?

Moving mesmo right after the verb changes its syntactic scope. Instead of emphasizing the subject ("I myself"), it now acts as a focus adverb modifying the verb, meaning "I really imitated the voice" or "I did imitate the voice".

Why use daquele here instead of desse to mean "that"?

Both mean "that", but daquele (a contraction of de + aquele) is used for someone distant from both the speaker and the listener in space or time. If the singer were near the listener, or if the listener had just brought the singer up in conversation, you would use desse.