Questions & Answers about Mi hermano es muy responsable.
Since this lesson is about ser and estar, could I say está muy responsable if he is just acting responsibly today?
No, responsable is an adjective almost exclusively used with ser. If you want to say someone is acting responsibly at this exact moment, you wouldn't use estar responsable. Instead, you'd use an active phrase like está actuando con responsabilidad (he is acting with responsibility) or se está comportando de manera responsable (he is behaving responsibly).
How would I say my brother is responsible for something, like a project or a pet?
You still use ser, but you add the preposition de rather than por. For example, you would say Mi hermano es responsable del proyecto (My brother is responsible for the project). Often, Spanish speakers turn it into a noun phrase to mean 'the one responsible': Mi hermano es el responsable del proyecto.
If I were talking about my sister, would the word change to responsabla?
No, adjectives that end in -e in Spanish, like responsable, inteligente, or amable, are gender-neutral. They stay the exact same whether you are talking about a man or a woman, so you would say Mi hermana es muy responsable. You only change them to match plural nouns by adding an -s (mis hermanos son responsables).