Я хочу нарисовать красивый пейзаж.

This is about verb aspect, which is very important in Russian.

• рисовать = imperfective, focusing on the process, repetition, or ongoing action
• нарисовать = perfective, focusing on completing the action

In Я хочу нарисовать красивый пейзаж, the speaker means I want to create/finish one picture. That is why нарисовать is natural here.

Compare:

• Я хочу нарисовать пейзаж. = I want to draw/paint a landscape.
  
  • emphasis on completing it
• Я хочу рисовать пейзажи. = I want to draw/paint landscapes.
  
  • emphasis on the activity in general

So here, нарисовать fits because the sentence refers to one finished landscape.

Because красивый must agree with пейзаж in gender, number, and case.

Here:

• пейзаж is masculine
• it is singular
• it is in the accusative case because it is the direct object of нарисовать

For masculine inanimate nouns, the accusative looks the same as the nominative. So:

• nominative: красивый пейзаж
• accusative: красивый пейзаж

That is why the adjective stays красивый.

Пейзаж is in the accusative case.

It is accusative because it is the direct object of the verb нарисовать: it is the thing the speaker wants to draw.

• Я хочу нарисовать что? → пейзаж

A useful thing to remember:

• masculine inanimate nouns often have the same form in nominative and accusative

So:

• nominative: пейзаж
• accusative: пейзаж

The case is accusative, even though the form does not change.

Because Russian does not use a separate word like English to before the infinitive.

In English:

• I want to draw

In Russian:

• Я хочу нарисовать

The infinitive form itself already shows that the verb is to draw / to paint.

So нарисовать by itself means to draw / to paint.

Yes, very often.

Russian often drops subject pronouns when the meaning is clear from the verb form. Since хочу already means I want, you can say:

• Хочу нарисовать красивый пейзаж.

This still clearly means I want to draw a beautiful landscape.

Including Я is also correct. It can sound:

• neutral
• slightly more explicit
• sometimes a bit more emphatic, depending on context

So both are fine:

• Я хочу нарисовать красивый пейзаж.
• Хочу нарисовать красивый пейзаж.

No, Russian word order is fairly flexible.

The most neutral order here is:

• Я хочу нарисовать красивый пейзаж.

But you could change the order for emphasis, for example:

• Красивый пейзаж я хочу нарисовать.
  Emphasis on beautiful landscape
• Нарисовать красивый пейзаж я хочу.
  More marked, poetic, or contrastive

Even though Russian allows movement, beginners should usually stick to the neutral order unless they have a reason to emphasize something.

Хочу means I want. It comes from the verb хотеть = to want.

This is the 1st person singular form:

• я хочу = I want
• ты хочешь = you want
• он/она хочет = he/she wants

It is not formed in a completely regular way, so it is worth memorizing.

It can mean either draw or paint, depending on context.

Russian рисовать / нарисовать is broader than English draw in some situations. It can refer to making a picture in general.

So:

• нарисовать пейзаж can mean draw a landscape
• and in many contexts, it can also be understood as paint a landscape

If you want to be very specific about painting with paints, Russian may sometimes use other expressions too, but нарисовать пейзаж is perfectly normal.

It actually is in the correct case already; it just happens that the form does not visibly change.

Пейзаж is:

• masculine
• singular
• inanimate

For many nouns like that, the accusative singular is identical to the nominative singular.

So you get:

• nominative: пейзаж
• accusative: пейзаж

That is why it may look like nothing changed, even though grammatically it did.

The stress is:

• Я хочУ нарисовАть красИвый пейзАж.

More clearly:

• я — unstressed or lightly stressed
• хочУ
• нарисовАть
• красИвый
• пейзАж

Stress matters in Russian, so it is good to learn words with stress from the beginning.

Russian has no articles.

English says:

• a beautiful landscape
• the beautiful landscape

Russian does not use separate words for a/an/the. So:

• красивый пейзаж can mean a beautiful landscape
• or the beautiful landscape

The exact meaning depends on context.

In this sentence, English would usually translate it as a beautiful landscape.

Yes. Красивый is a common general adjective meaning:

• beautiful
• pretty
• handsome
• lovely

The best English translation depends on context.

With пейзаж, beautiful is the most natural translation:

• красивый пейзаж = a beautiful landscape

Yes, but it changes the nuance.

• Я хочу нарисовать красивый пейзаж.
  = I want to draw/paint a beautiful landscape.
  Focus on finishing it.

• Я хочу рисовать красивый пейзаж.
  = I want to be drawing/painting a beautiful landscape.
  Focus on the process or activity.

So the second version is possible, but it does not sound exactly the same. In your sentence, the perfective нарисовать is the more natural choice if the speaker means creating a complete picture.