Are なんて and なんか used to emphazise feelings that are not disgust/trying to look down on something in this context?

Question

I know that なんて and なんか are used to emphazise disgust, belittle something, say that something is undesirable and express negative feelings of the speaker. But in this context I'm not too sure if this applies...

Context: girl is talking with her boyfriend over the phone. She looks very happy and also nervous while he's telling her about his day. But her voice sounds too monotone/inexpressive, which makes her boyfriend think that she doesn't actually care about what's he saying. He tells her that she sounds like she doesn't care at all because she sounds too monotone and calm. However, it's important to say that he's not angry at all, he's just teasing her and giggling while telling her she doesn't care. Then she replies saying

そんなことないですよ... 聞いてますよ.

But he keeps teasing her about her not listening, so she says

聞いてますよ! どうでもよくなんかないよ. While in her mind she thinks

"余裕なんてない"...

Again, keep in mind that she says all of this without getting angry at all over his teasing or anything. She's just smiling, and in fact looks pretty happy and a bit anxious. That's why I'm not sure what's the function of なんて and なんか.

どうでもよくなんかないよ. (It's not that I don't care)

余裕なんてない = I'm not calm at all (when I talk to you)

I do notice that both sentences are in negative, so the word choice is fine, but like I said, since the girl is happy while using these words, I'm not sure if なんて and なんか are trying to emphazise or belittle 余裕 and どうでもよく or why they're being used in this context if she doesn't mean to emphazise/express disgust.

Answer 1

What you learned, that なんて and なんか emphasize the negative feelings of the speaker towards the mentioned, is certainly a (most?) outstanding usage of the words, but at the same time the furthest stretched out definition of them. Their etymological meaning is something like "such kind of thing", and there exist a wide gradation of meanings between the literal one and the said one, which cognitive linguists often refer to as polysemy network.

• 野球なんか／なんてしながら過ごす "spend (time) playing baseball and such things" (neutral)
  • Positive branch
    • 野球なんか／なんてどうですか？ "Why don't we do something, namely, baseball?" (suggestion, recommendation)
    • へえ、野球なんてするんだ！ "Wow, you do play (such thing like) baseball!"(astonishment, admiration)
  • Negative branch
    • 野球なんか／なんて小学生以来です "Baseball... I haven't played any since elementary school" (denial of existence)
    • 野球なんか／なんてしてる場合じゃないぞ！ "This is not a time to play baseball at all! (don't you see?)" (objection, opposition)
    • 野球なんか／なんてつまらない "baseball is just so dull" (repulsion, belittling)

The classification shown above is nothing authoritative. I only hope you to grasp the general idea of the semantic development. So, back in your original question, what those なんか and なんて in her speech represent? It seems to me in this case, a mild nuance of objection alongside negation, or to say: no, you're wrong; it's not like that at all.