Я хочу посмотреть на звёзды с балкона.

Because посмотреть is the perfective form, and here it means something like have a look or look for a while.

After хочу, Russian can use either aspect, but the choice changes the nuance:

• Я хочу посмотреть на звёзды = I want to look at the stars, as a complete action or for a bit
• Я хочу смотреть на звёзды = I want to be looking at the stars, emphasizing the process or duration

In this sentence, посмотреть sounds very natural because the speaker means a simple intended action.

In this verb, по- often gives the idea of doing something for a short time or as a single completed action.

So:

• смотреть = to watch, to look
• посмотреть = to look, to take a look, to watch and complete that action

It does not always mean exactly a little, but that is often a helpful way for learners to feel the difference.

Because the verb смотреть / посмотреть usually takes the preposition на when it means look at.

So Russian says:

• смотреть на небо = look at the sky
• смотреть на море = look at the sea
• смотреть на звёзды = look at the stars

Without на, the phrase would not sound normal with this meaning.

Because на after смотреть / посмотреть requires the accusative case.

The noun is звезда. Its plural forms are:

• nominative plural: звёзды
• accusative plural: звёзды

For this noun, the nominative plural and accusative plural look the same, because it is an inanimate plural noun in this context.

So:

• звёзды = stars
• на звёзды = at the stars

Because с here means from, and it expresses the place you are looking from.

So с балкона = from the balcony.

The noun балкон changes to балкона because с in this meaning takes the genitive case:

• балкон = balcony
• с балкона = from the balcony

Yes, and the meaning is close, but the nuance is different.

• с балкона = from the balcony, focusing on the viewpoint
• на балконе = on the balcony, focusing more on location

So:

• Я хочу посмотреть на звёзды с балкона = I want to look at the stars from the balcony
• Я хочу посмотреть на звёзды на балконе = I want to look at the stars while on the balcony

In many situations, both are possible, but с балкона is especially natural when you mean the balcony is your vantage point.

Yes, often you can.

Russian often drops subject pronouns when the verb already makes the subject clear:

• Хочу посмотреть на звёзды с балкона.

This still clearly means I want to look at the stars from the balcony, because хочу is the I form.

Including Я is also correct. It can sound a bit more explicit or emphatic.

Yes. Russian word order is flexible, and different orders can shift emphasis.

Neutral order:

• Я хочу посмотреть на звёзды с балкона.

Other possible orders:

• С балкона я хочу посмотреть на звёзды.
  Emphasizes from the balcony
• На звёзды я хочу посмотреть с балкона.
  Emphasizes at the stars
• Я с балкона хочу посмотреть на звёзды.
  Also possible, with slightly different focus

The original sentence is a very natural, standard order.

Because Russian expresses want to do something by using the verb хотеть plus an infinitive.

So:

• я хочу спать = I want to sleep
• я хочу читать = I want to read
• я хочу посмотреть = I want to look

This is similar to English want to + verb, except Russian does not use a separate word for to before the infinitive.

Yes. In normal Russian writing, ё is often written as е, so you may see:

• звёзды
• звезды

Both represent the same word, but звёзды is more precise because it shows the correct pronunciation and stress.

The stress is on ё:

• звЁзды

That is useful for learners, because stress matters in Russian.

A rough pronunciation guide is:

• ya kha-CHOO pa-sa-MO-tryet na ZVYOZ-dy s bal-KO-na

A few tricky points:

• х is like the ch in Scottish loch or German Bach
• посмотреть has stress on -треть: посмотрЕть
• звёзды has stress on звЁ-
• балкона has stress on -ко-: балкОна

If you want to sound natural, do not pronounce every word too strongly. Russian unstressed vowels get reduced quite a lot in normal speech.