: First conversation scenes I've written (looking for errors, conventions, and improvements according to writing standards) I would like to know if I'm doing something wrong (according to writing

I'm just rewriting. You need more narration and action tags:

She stood in front of his desk for a full minute before he noticed her. When he looked up at last, his only greeting was a raised eyebrow.

“Uhm...excuse me, are you Dr. Aide?”

"Speak up. You're muttering."

She cleared her throat twice. "Are you Dr. Aide?"

“Yes.”

“My...my name is Luna, I a new nurse here. I'm — I'm going to be your assistant. As of tomorrow." When he didn't respond, she added, "If that's all right?"

“I see.” he replied. She looked down at her hands, clutching the folders in front of her like a shield. "Please take a seat," he continued, surprising her.

She pulled out the wooden chair, wincing as it scraped on the linoleum, and sat, her arms still wrapped around the case folders. The doctor reached down and opened a desk drawer, producing a bottle of vodka. Her eyes widened as he topped off what she had taken for a water glass. There was a second, empty glass beside it. He motioned the bottle towards her and raised an eyebrow again.

Drinking on the job... what kind of man is this? She wet her lips. He's probably testing me.

"Okay, but... just a little."

He filled the glass halfway and pushed it towards her. She took it, proud to see that her hand only shook a little. The vodka was unexpectedly smooth, but she didn't trust herself with more than a sip. Dr. Aide took two healthy slugs.

The silence lengthened between them. Luna began to feel desperate, wondering if she was failing the test. "Do you always stay so late, Doctor?" she ventured.

His mouth twisted up in a smirk. "I'm here almost 24 hours a day."

“You must love your job.”

“Not really.” His eyes swept the papers, the antiquated computer, the dusty file cabinets, the hospital-green paint peeling from the cinderblock walls. “Actually, I’ve been thinking of quitting.”

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You are using adverbs and to tell us how the characters are acting and feeling. Instead consider showing us their body language. Help me hear them speak. Let me see for myself that Luna is nervous and Dr. Aide is worn out, jaded, or whatever it is that he is.
From a grammar perspective (although those questions are best asked on English.SE), a sentance in quotations that ends in a period and is before "he says" or "she says" or the like is usually ended with a comma, not a period.
I'm curious why everything is in the present instead of the past tense. I am much more used to reading "he said" and "she said." That doesn't mean that there's anything wrong with present tense in fiction, but you should look into the effect it has.
Finally, don't overpunctuate within the dialogue. Consider leaving out ellipses, for example. Instead break it up with your character attributions ("he says") or through showing me the pauses by the action. The punctuation can become visual clutter.
I've made some minor changes, but not necessarily incorporated all of my suggestions, below.

“Uhm, excuse me, are you Dr. Aide?”
she asks. Her voice is soft and
tentative.
“Yes,” he replies.
“My...my name is Luna, I a new nurse here. I
will assist you from tomorrow,” she
says.
“I see,” he replies, “Please
take a seat.”
She sits in front of him
feeling a bit awkward. He waves his
bottle of vodka. “Do you wanna some?”
“OK," she says, "just a little.” He
fills the glass halfway They drink in
silence. The seconds fill like
minutes. Luna breaks it. “Do you
always stay so late Dr. Aide?.”
“Yes,
I spend almost 24 hours in this
place.”
“You must love your job.”
“Not
really,” he says, then pauses. “Actually I’ve been
thinking of quitting.”

Here you do not follow a question mark directly with a period. It stands on its own.

“So what’s your definition of a mental illness?” he asks.
She thinks for a while and then replies, “someone who can’t adapt himself to society.”
He looks out of the window and says, “It is no measure of health to be well adjusted to a profoundly sick society. Jiddu Krishnamurti.”
“What does that mean?” she asks.
“What if a society is sick already by itself? Then we need another meaning. I think it is any behaviour that threatens humanity and the ecosystem in direct and indirect ways.” he replies.

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