This function of this か is not purely phonetic, but rather serves to make the sentence less of a outward statement and more of a self-directed or self-reflecting one.
It makes the information value of sentence primarily be “I had considered ~ previously but wasn’t sure, but in the end it indeed it is 〜, huh...”
It’s often is accompanied by やっぱり (or 予想通り ...
This kind of interrupting parentheses are a common practice in Japanese too. Beware that, however, the linguistic and cultural difference between these two languages might make literal translations not working.
As for your examples,
The parentheses seems hardly useful to me, partly because this 祝日 is felt neither exceptional (...
Basically かも is treated like one word, and there should be no pause between か and も. If you stopped talking after か for whatever reason, you should repeat at least かも. As long as you treat かも as one word, it's possible to continue the other person's statement and say something like this.
"Is it expensive?"
Because speaking casually very often means discarding the question mark か as well, making the casual version of そうなんですか？ -> そうなんだ？
It's just not そうなんだか, but "だか" that you wouldn't hear in any sentence, because you would simply stop at だ。
Both can be a negative rhetorical question, but are used in different situations.
この車は新しくないんですか is "Isn't this car new?" It sounds like they are talking about the age of one specific car. More literally, "As for this car, isn't it new?"
この車は新しいんじゃないですか is "Isn't this one a new car?" It sounds like they are finding a new car ...