Aww! Not another one! We haven't even learned the word for woman yet!
How do I know if an adjective takes -í or -y as an ending when it describes a male noun? Is it random, meaning does it have to be memorized when learning the words, or is there a rule?
Not exactly random. There are hard and soft adjectives, and the soft adjectives take -í regardless of gender. You do have to learn which adjectives are soft, but it gets easier when you hear phrases like další žena.
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.
Is 'muž' refering to a male person in general or would i use it for 'husband'?
"muž" means a male person in general, but when you use "můj muž" (literally translation - "my man") it means "my husband". In the same way "její muž" means "her husband".
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].