Yes, everything you just posted works without problems. Hopefully this will clear things up.
When you say
静かだ you are literally saying
is quiet in English. The same can be said of い adjectives. Ending a sentence with
大きい really just says
is big. With adjectives, the to be verb is already built in. With い adjectives saying です after the adjective, while grammatically correct is not always necessary. You would add です to be polite to the listener.
When you attach the adjective to a noun,
静かな人 you no longer have the to be verb as a closer. In fact, in these situations, you can safely assume that there is no verb, and that the adjective + noun form a noun clause. In short, you need a verb, and だ/です will work for you. The same will apply to い adjectives as well.