"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
"是……的" is an emphasis pattern and also the most natural way to provide certain information, but cafe_disco is also correct about the different patterns for linking an adjective to a noun.
The following article (and its further links) touches on most of the uses of "是……的", though it doesn't really address its use for simple adjectives, which can indeed be to emphasis them, but I think the thing to take from cafe_disco's comment is that when putting them around an adjective "是" doesn't work on its own — it goes with "的" (which, in contrast, can in fact stand on its own).
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.