Learn Spanish (Latin America): Habiendo analizado el testimonio, el juez escribió el veredicto. - Having analyzed the testimony, the judge wrote the verdict.

You use habiendo analizado (having analyzed) because the analysis was completely finished before the judge wrote the verdict. If you used the regular gerund analizando (analyzing), it would imply the judge wrote the verdict while analyzing the testimony at the exact same time.

Can I rephrase the beginning of this sentence to use después de?

Yes! A very common and slightly less formal way to say this is Después de analizar el testimonio... (After analyzing the testimony...). Notice that after prepositions like después de, Spanish uses the infinitive (analizar) rather than a gerund form.

Why is the main verb escribió in the preterite rather than the imperfect (escribía)?

The preterite escribió is used because writing the verdict is viewed as a single, completed event in a sequence of actions (first the judge analyzed the testimony, then wrote the verdict). The imperfect escribía would sound like the judge was in the middle of writing it when something else interrupted.

If the judge were a woman, would it be la juez or la jueza?