First, people do not notice "he said" or "she said", and you should use them more often. We expect them, they work, and they are not intrusive; stop thinking they are.
That said: When there are only two people in a conversation, attributions are seldom necessary unless your break the repeated pattern of "One talks, the other talks". Which realistically you might do; sometimes people don't answer, or pause before answering, or sometimes people finish a thought and have an afterthought and voice that.
Absent those special circumstances, put in a "said" every four or five lines of dialogue just to keep the reader oriented.
The practice of combining dialogue with action (as you have done) is alright, but is overdone when the action is superfluous, neither necessary or natural, which seems to be the case in several of the spots of your writing. The actions you are describing do not add anything to the story, in my opinion, and because of that it becomes noticeable you are "fluffing it up" for no purpose. (Well your purpose is to avoid "said" tags, but that is not a valid purpose from the reader's POV, because the tags are nearly invisible to them and professional authors use them all the time.)
IMO Trying to lay all your exposition and back-story in-between lines of dialogue is awkward and causes me to lose track of the conversation altogether.
Exposition isn't bad, it is also part of story telling. The trick is to keep it short, a few lines or a paragraph. It only stalls the story when it starts to dominate a page. Everything you have to say about Noah and his history could be said in a few lines.
As far as the "general standards" of good dialogue, I would say the best dialogue contains conflict, at least for the POV character. This can be banter, or somebody doesn't want to believe what they are being told, or an actual argument, or romantic or social tension.
I think you are aiming for romantic/social tension, that Emmy is attracted to Noah. The conflict is a result of him being either oblivious to her feelings or incapable of returning them, and also a result of her being unwilling to be explicit about those feelings, and approaching them more indirectly, with pretenses.
That is a fine dynamic. But all these places where you use actions to introduce statements, where these actions are meaningless, you could instead be giving us information on what she actually feels or thinks or imagines happening with Noah.
Bring back the "said" tags and do that instead. She can be honest in her own head, and instead of the action tags (and backstory) distracting from the conversation, her thinking about what she feels and is trying to accomplish here will raise the stakes on this conversation, and keep us glued to the conversation happening.
It's the iceberg analogy. What is actually verbalized in a conversation is 10% of the communication; the other 90% for Emma is in her feelings, thoughts, reactions and struggle to make a connection with a boy she wants that happens to not know how to do his half of the work.
For Noah, even though you are not showing his thoughts, the other 90% of this conversation is insecurity, confusion, and social paralysis, because (as you convey by the birthday wishes) he also would like to make this connection, but doesn't want to embarrass himself and ruin any chance of it by blurting out his own feelings. So he is stuck in conversational paralysis, his mind racing but afraid to say anything that might repel or repulse her. So all he's got is banalities.
The birthday wishes is what Emmy should truly pick up on, if she has any smarts at all. Not wandering off into her back story so nothing happens with those wishes. Noah is thinking about her, and personally. That means he's NOT really off in his own world and only talking to her when she happens to show up, he wants at least some of this connection with her too.
That's a key, it should make her feel something good and momentous, and perhaps it unlocks her own reticence and inexperience and gives her the courage to initiate the next step forward with Noah, whatever that might be. Then the birthday wishes mean something to the plot here, they aren't just a throwaway line.
As for the reason she doesn't celebrate birthdays, that is for dialogue later, a personal reveal to somebody in the future. That can have a plot implication too. If it is Noah, much of what we feel about other people is by sharing intimate hardships most others don't know about. Just about everybody is willing to share victories and happy times with even casual acquaintances, but (for most people) you have to be particularly close to somebody to share your heartbreaks with them. We know that instinctively, so when somebody we like and have become friends with does share a heartbreak with us, it makes us feel closer to them, it is a kind of non-sexual intimate moment. You can use that in your fiction, to bring Emmy closer together with Noah (or anybody).
Now I realize that is probably too much for a 1000 word story. But I am adding it as a principle of good dialogue, as you requested.
Dialogue should change hearts and minds (feelings or thinking) of at least one character. That change should be connected to the plot and mean something to the story. It is not a place to shove backstory or other exposition.