Learn French: Le chercheur est fatigué, alors il va dormir. - The researcher is tired, so he is going to sleep.

Yes, you could completely say Le chercheur est fatigué, donc il va dormir. Both words express a logical consequence. Alors translates well to "so" and is very common in everyday spoken French. Donc is closer to "therefore" or a slightly stronger "so," making the logical link a bit more formal or obvious.

If the researcher were a woman, how would the sentence change?

It would become La chercheuse est fatiguée, alors elle va dormir. The masculine noun chercheur changes to chercheuse, the pronoun il changes to elle, and you must add an e to the adjective fatigué to make it agree with the feminine subject.

Why is it va dormir instead of just dort?

Il va dormir means "he is going to sleep," which is the near future tense. If you used the present tense il dort, the sentence would mean "The researcher is tired, so he is sleeping." This near future tense is formed exactly like it is in English: the present tense of aller (to go) plus the infinitive verb dormir (to sleep).

Does alors have any other meanings I should look out for?