(1) what does the なった means here?
気分 is "feeling, mood". According to the provided context そんな気分 is just pointing to the feeling stated in the first sentence (my attempted translation might not be accurate):
I forgot it after waking up, but something is missing
Therefore, そんな気分 would be "The feeling of finding that something is missing".
I think you are correct than なった is the past verb to mean "became", and the construction here is A + に + なる (where A is a noun or a na-adjective) which means "to become A" :
I got cold.
[She] became beautiful.
In your sentence, I understand it as though the speaker didn't have this feeling initially, but at some point he got it.
~のは as ending, how does it work/what nuance does it give to the sentence?
I don't think that のは is meant to be a sentence ending construction. Note that it is not the case that your sentence ends in a period:
But instead, if we look at the context, it is followed by the ellipsis "...", and after that by a "たぶん...":
そんな気分になったのは ...、たぶん ... [something implied or deferred as an ending]
which suggests that grammatically it should have an ending. the の is nominalizing the predicate そん気分になった (please take a look to this Q&A for more info on nominalizing sentences with の) and the は is making the nominalized predicate the topic of the incomplete sentence:
そんな気分になったのは... [something missing]
Getting this feeling... [something missing]
There is a grammar construction Aのは、Bからです meaning "the reason for A is B":
The reason it has become cold is that the window is open.
Note that in this construction, the reason A is being emphasized by the use of のは, so it sounds stronger than just:
This is the grammar I guess is being used in your sentence. Maybe I am wrong and のは is just emphazising the sentence as a subject without being linked to the reasons for it.
All in all, my guess is that leaving そんな気分になったのは ...、たぶん ... as an incomplete sentence is a narrative tool, and that what comes after the sentence is the exposition of the events or the facts that explain why the speaker had such a feeling.
Somehow, if this was a movie or an anime, it would be the equivalent to a narrator voice explaining something, and then changing the focus to a scene, a dialog or a flashback where the narrator shuts up and the characters start interacting. So all in all, I would understand it as:
そんな気分になったのは... たぶん... [Events shown directly in the story]
The reason why I got that feeling is, maybe... [Events shown directly in the story]