How do I identify which would be a hard or soft adjective though? I don't know what that means really. Also, you're saying that gender doesn't matter or masculine vs feminine doesn't matter. (If you by chance have learned Spanish before and can compare it to Czech with this topic that would literally be amazing if it relates and has similar rules, but I'm not expecting it to since they are such different languages lol)
It doesn't "mean" anything really, it's just a class of words with the same grammatical rules. You will have to learn to recognize them or memorize it. The name "soft" comes from the fact that their ending contains the letter "í" (also called soft í in Czech), and generally they usually sound softer. The (grammatical) gender does not matter for the soft adjectives in nominative. It makes a difference in other cases and also for the hard adjectives. You can read some of this in the notes to this lesson.
In English usage we generally need either "another man" or something like "the/that/this/some other man," depending on context. Since there is neither context nor a demonstrative in the Czech sentence, I'm not sure the last ones fit the exercise. But I will add those, if any, that the Czech natives on the team feel would be appropriate here.
No, they are different, but please see the Tips and notes for the first skill https://www.duolingo.com/skill/cs/Phrases/tips-and-notes or any other introductory material.
What often happens is the assimilation in the voiced-unvoiced pairs. If a voiced consonant stays alone at the end of the word like this, it is produced unvoiced. So ž->š, d->t, v->f and similar.
So, to be explicit, muž is pronounced [muš] and muže is pronounced [muže].