I'll let dad know about Judy!!”

You could also say "I'll warn dad about Judy!" The way you have it is correct, and I don't think it sounds too formal, but you could change it like the other person said if you wanted to.

“That's impossible.”,” Alice's mom said calmly, "you can't see your dad anymore. You can't get out of here."

Like the other commentor showed, these next few sentences are better as one quote. I hope my explanation makes sense.

This structure is common in stories or books written in English:
"A," said a character, "B."
where the sentence(s) are split around the description of who is speaking.

I think you were close with "... said calmly and continued," but if you use commas and make it one quote, you don't need to say "and continued" explicitly.

Alice's mom said calmly and continued.,

“Yyou can't see your dad any more.

You can't get out ofrom here.”

You could also say:
"You can't escape from here." or
"You can't go back."

Or even, "We can't get out of here," because everyone is also trapped here.

“Why not?

I think you want to show Alice asking "why" her mom is saying she can(not) get out. So, she should say "Why not?"

I can go back (to dad) when I return to where I came in from.”

I tried to keep what I think you are saying. You can remove "to dad" but it is okay to leave it too. If Alice is saying she just needs to leave "the way she came in" (go back to the door she opened), you can say "where I came in from."

"came in" is used for doors people have gone through. It's usually for when people come into a room or building from outside, but I think it makes sense here.

Alice turned to the waywards where she came alongfrom, but the door on the side of the tree trunk had vanished.

"to the way" sounds awkward. "towards" sounds more natural, and means almost the same thing.

When Alice looked at the tree, the door was already gone, so you need to say that the door "had vanished." "had" is the past tense of "to be."

VIt vanished!!”

In English you need to include a subject -- even if it's just "it," otherwise it isn't a complete sentence.

I don't think "vanished" is too formal, but you could also say "It disappeared!" or "It's gone!" if you like.

She shouted uneasily.

If you want to apply an adjective (uneasy) to a verb (shouted), you need to use an adverb. In this case, you can use "uneasily."

“Alice, you've been are already dead”

Because the mother is telling Alice this now, and the condition "Alice is dead" is still happening when the reader reads this, you should use present tense. Also, when you use "already" I think you usually use a present tense.

Alice, you have been ... >> Alice you (are)

“WhyHow?

"How?" or "What?" makes more sense here.

Am II am dead?

I'm sure this is a confusing correction. What you wrote IS the correct structure for a question, but Alice isn't really asking "Am I dead?" She is just shocked to hear this and asks in disbelief. She repeats what her mother says, but with a question tone, because she is surprised.

So you should keep a "statement" form instead of changing to a "question form" with "to be" at the beginning. Examples of what I'm talking about:
statement form: I + am + dead. (subject + "to be" + verb)
question form: Am + I + dead? ("to be" + subject + verb) asking for Yes/No answer.

TheHer mother said crying.

“Your dad will come here some time soon.

Let's live here together with the three of us.”

Alice realized that (all) the people here were dead ones(already) dead.

You don't need to include "ones," as "dead" is a verb and complete the sentence. You could leave out "all" if you want to, or add "already" before "dead."

Judy was happily talking to Alice's father.

This sentence is fine, I just think the transition is sudden. I would either add a separation (an empty line) or a sentence to show the perspective (think of it like a "camera") changed to focus on Judy and Alice's dad.

Did something nice happen to you?”

"to you" isn't wrong, but it is more natural to leave this out.

“Ah, I'm just tipsy because of the wine.”

The noun "wine" needs an article before it, and because she is talking about the specific wine they drank with the meal, you should use the definite article "the."

She continued to talk about something to him, happily laughing.

You don't need to include "about something," and it sounds more natural to just say that she talked to him.

Her faint laughter faintly reached to Alice's room.

She put hHer face was down on the desk and she was bleededing from her mouth.

If you are describing the condition that the "camera" finds Alice in, you shouldn't use "She put..." because it sounds like she is conscious and just now is putting her face down on the desk.

I think because Alice was talking to her mother in Wonderland, that she was already dead, and the "camera" is just finding her body in the real world now.

I used continuous tense for "her face was..." and "... she was bleeding..." so they matched and sounded natural.

The heart-shaped pendant faromund her neck was beautifully gleaming.

I'll let dad knowtell dad about Judy!!”

Grammatically your original sentence is correct, but I think in casual speech it would always be phrased as "I'll tell dad" rather than "I'll let dad know" (which sounds too formal).

“That's impossible.,”

Alice's mom said calmly and continued.

You can't get out ofrom here.”

I can go back to dad when I return togo back where I came from”

Again, your original sentence is grammatically correct, but the word 'return' is a bit too formal for casual speech.

Alice turned to face the way she had caome along, but the door on the side of the tree trunk had vanished.

VanishedIt's gone!!”

Again, vanished is the right word for what you mean, but it's not used in casual speech.

She shouted uneasy, disturbed.

To me, being uneasy is a very low-key emotion, so one can't really shout and be uneasy at the same time. Shouting calls for a stronger emotion.

“Alice, you've beenre already dead,”

Hher mother said softly.

“Whyat?

Am II am dead?

“The rabbit is a guide for dead souls.,”

Ther mother said, crying.

“Your dad will come here some time soon.

Let's live here together with-- the three of us.”

Alice realized that all the people here were dead ones.

“Ah, I'm just tipsy because offrom the wine.”

She continued to talk about something to him, happily laughing.

If it's unclear/unimportant what they're talking about, you don't need to clarify "about something".

Her faint laughter faintly reached to Alice's room.,

Alicwhere she was sitting at her desk.

She put hHer face was down on the desk ands she bleeded from ther mouth.

The heart-shaped pendant hanging from her neck was beautifully gleaming.