Я не хочу испачкать новую рубашку соусом.

Because after хотеть / хочу Russian normally uses an infinitive to say what someone wants or does not want to do.

• Я хочу читать = I want to read.
• Я не хочу ждать = I do not want to wait.

So Я не хочу испачкать... literally means I do not want to stain / get ... dirty.

This is a question of aspect.

• испачкать is perfective
• пачкать is imperfective

Here, the speaker is talking about one unwanted result: ending up with the shirt stained. That is why испачкать fits well.

• не хочу испачкать рубашку = I do not want to get the shirt dirty
• не хочу пачкать рубашку = I do not want to be dirtying the shirt / I do not want to dirty the shirt in general

So the perfective verb focuses on the completed result: the shirt becoming stained.

Because it is the direct object of the verb испачкать.

The verb answers что? = what?

• испачкать что? → рубашку

Since рубашка is a feminine noun, its accusative singular form is рубашку.
The adjective must agree with it, so:

• новая рубашка = nominative
• новую рубашку = accusative

So both words change because the phrase is the object of the verb.

Because with испачкать, Russian often uses the instrumental case for the substance that causes the stain.

You can think of it as answering чем? = with what?

• испачкать соусом = stain with sauce
• испачкать грязью = stain with dirt
• испачкать краской = stain with paint

So соусом means with sauce in the sense of by means of sauce / using sauce as the staining substance.

Because the verb испачкать can take the instrumental case directly.

So Russian says:

• испачкать рубашку соусом

not

• испачкать рубашку с соусом

Here the case ending itself shows the meaning. Russian often does this instead of using a preposition.

Yes. In many situations, you can simply say:

• Не хочу испачкать новую рубашку соусом.

That works because хочу already shows first person singular: I want.

Including Я makes the subject more explicit and can add emphasis or contrast:

• Я не хочу испачкать новую рубашку соусом.
  = I do not want to stain the new shirt with sauce.

No, Russian word order is fairly flexible because the case endings show the grammatical roles.

The sentence as written is neutral and natural:

• Я не хочу испачкать новую рубашку соусом.

But you can move parts around for emphasis:

• Новую рубашку я не хочу испачкать соусом.
  Emphasis on the new shirt
• Соусом я не хочу испачкать новую рубашку.
  Emphasis on with sauce

So the original order is the most neutral one, but not the only possible one.

It usually implies avoiding an unwanted result, often something accidental.

So the feeling is not just I do not want to stain it on purpose, but more like:

• I do not want to end up staining it
• I do not want to accidentally get it dirty

That sense comes partly from the perfective verb испачкать, which focuses on the result.

Yes. Запачкать is very common and often very close in meaning.

• Я не хочу запачкать новую рубашку соусом.

This would sound natural too. In everyday speech, испачкать and запачкать often overlap a lot when talking about getting something dirty or stained.

Because Russian adjectives agree with the nouns they describe in:

• gender
• number
• case

Since рубашку is feminine singular accusative, the adjective must also be feminine singular accusative:

• новая рубашка = a new shirt
• новую рубашку = a new shirt, as the direct object

This kind of agreement happens all the time in Russian.

Because Russian has no articles.

So рубашку can mean:

• a shirt
• the shirt

The exact meaning depends on context. In this sentence, новую рубашку is most naturally understood as something like the new shirt or my new shirt, but Russian does not need a separate word like the to say that.

The main stresses are:

• хочу́
• испа́чкать
• но́вую
• руба́шку
• со́усом

So a rough pronunciation guide is:

ya nye kha-CHOO ees-PAHCH-kat' NO-vu-yu ru-BASH-ku SO-u-sam

A natural spoken rhythm would be:

Я не хочу́ испа́чкать но́вую руба́шку со́усом.