"Of course she is serious."
Yes, I believe so. I think that if you want to link a noun with an adjective, you can use any of the following structures:
noun + adjective
noun + 很 + adjective
noun + 是 + adjective + 的
I'm sure that each construction will give off a slightly different feel/connotation, but I'm not a native Chinese speaker, so I couldn't tell you what those would be. I hoped that helped somewhat
Sure. If you want to say "She is happy", you can just say 她高兴. I'm assuming, however, that it's more common to use the "noun + 很 + adjective" construction. Even though 很 is technically defined as "very", in actuality it carries very little weight, and usually just links the noun to the adjective in a neutral manner.
Again, I'm not a native Chinese speaker, but this is what I've gathered from studying the language.