No quiero salir del coche hasta que deje de llover.

Because de + el contracts to del in Spanish.

So:

• de el coche → del coche

This is a standard contraction you must use.
A similar one is:

• a + el → al

So salir del coche means to get out of the car.

In Spain, coche is the normal word for car.

In much of Latin America, people often say:

• carro
• auto
• automóvil

Since this is Spanish from Spain, coche sounds the most natural.

Spanish often leaves out subject pronouns because the verb ending already shows who the subject is.

• quiero = I want

So No quiero already means I don’t want.
You could say Yo no quiero, but that usually adds emphasis, contrast, or emotion.

For example:

• Yo no quiero salir = I don’t want to go out / I’m the one who doesn’t want to go out

Salir de means to go out of or to leave a place.

So:

• salir del coche = to get out of the car

This is a very natural way to express that idea in Spanish.

You may also hear:

• bajar del coche

In Spain, bajarse del coche or bajar del coche is also common for getting out of a car, but salir del coche is completely correct and natural.

Hasta que means until when it is followed by a clause with a verb.

Compare:

• hasta las cinco = until five
• hasta mañana = until tomorrow
• hasta que deje de llover = until it stops raining

You use hasta que because what follows is a full clause: deje de llover.

Because after hasta que, Spanish uses the subjunctive when the action has not happened yet and is in the future from the speaker’s point of view.

Here, the rain has not stopped yet. The speaker is waiting for that future moment.

So:

• hasta que deje de llover = until it stops raining

Not:

• hasta que deja de llover

Deja is indicative, but here Spanish prefers the subjunctive: deje.

Dejar de + infinitive means to stop doing something.

So:

• dejar de llover = to stop raining

Other examples:

• dejar de fumar = to stop smoking
• dejar de hablar = to stop talking
• dejar de correr = to stop running

In your sentence, deje de llover literally means it stops raining.

Because it comes after dejar de, and after that structure you use an infinitive.

• dejar de + infinitive

So:

• dejar de llover
• dejar de comer
• dejar de esperar

Here, llover is the action that will stop.

Llover is an impersonal verb. Weather verbs often work this way in Spanish.

So Spanish simply says:

• Llueve = It’s raining
• Dejó de llover = It stopped raining

There is no real subject like a person doing the action.
English uses it as a dummy subject, but Spanish usually does not need one.

Spanish often uses the present tense in the main clause to talk about a current attitude or intention, even if the second action is future.

• No quiero salir... = I don’t want to get out... right now
• hasta que deje de llover = until it stops raining in the future

So the sentence combines:

• a present feeling or decision: No quiero
• a future event expressed with hasta que + subjunctive: deje de llover

This is completely normal in Spanish.

Yes. That is also correct and very natural.

• dejar de llover
• parar de llover

Both can mean to stop raining.

In many contexts, hasta que pare de llover may even sound slightly more everyday to some speakers, but hasta que deje de llover is perfectly standard and natural too.

Literally, it means I don’t want to.

• No quiero salir del coche = I don’t want to get out of the car

Depending on context, it can imply refusal, so in some situations English might translate it as something like I’m not going to get out of the car, but the basic meaning is about wanting, not simple future refusal.

Yes, that is possible, but get out of the car is the most natural meaning here.

Because of salir del coche, the idea is specifically going out of the car, not just leaving it in a more general sense.

So the best natural translation is:

• I don’t want to get out of the car until it stops raining.