Ich ziehe das gewaschene Hemd an.

Because the verb here is anziehen, which is a separable verb.

In a normal main clause:

• the conjugated part goes in the usual verb position
• the prefix goes to the end

So:

• anziehen → infinitive
• Ich ziehe ... an. → present tense main clause

A few related forms:

• Ich ziehe das Hemd an.
• Ich habe das Hemd angezogen.
• ..., weil ich das Hemd anziehe.

So the split only happens in certain sentence types.

Not usually. By itself, ziehen most often means things like pull, draw, or move.

The meaning put on clothes comes from the full verb anziehen.

So:

• ziehen = pull / draw
• anziehen = put on, get dressed, wear in some contexts, and a few other meanings depending on context

German prefixes often change the meaning of a verb quite a bit.

Das gewaschene Hemd is the direct object of the sentence, so it is in the accusative case.

The noun Hemd is neuter:

• nominative singular: das Hemd
• accusative singular: das Hemd

For neuter nouns with the definite article, nominative and accusative look the same, so you do not see a change in das.

That is why it stays:

• das Hemd

Because gewaschen is being used like an adjective before a noun, and adjectives in German take endings.

Here the noun phrase is:

• das gewaschene Hemd

Since:

• Hemd is neuter
• it is singular
• it has the definite article das
• it is accusative

the adjective takes the weak ending -e:

• gewaschene

Compare:

• das gewaschene Hemd = the washed shirt
• ein gewaschenes Hemd = a washed shirt

After das, the adjective gets -e. After ein, it gets -es.

It is originally a past participle of the verb waschen → gewaschen, but here it is being used as an adjective.

So in this sentence, it describes the shirt:

• the washed shirt

This is very common in German. Other examples:

• die geschlossene Tür = the closed door
• der gekochte Reis = the cooked rice
• das geöffnete Fenster = the opened/open window

So grammatically, it behaves like an adjective here, even though it comes from a verb.

Good question. Both versions are possible, but they are used a little differently.

• Ich ziehe das gewaschene Hemd an.
  = I put on the washed shirt.

• Ich ziehe mir das gewaschene Hemd an.
  = I put the washed shirt on myself.

The mir is a dative pronoun and often appears when talking about putting clothing on part of oneself. It can sound very natural in everyday German.

Also compare:

• Ich ziehe mich an. = I am getting dressed.

So:

• das Hemd anziehen = put on the shirt
• mir das Hemd anziehen = put the shirt on myself
• mich anziehen = get dressed

It can mean either one. German present tense often covers both:

• I put on the washed shirt
• I am putting on the washed shirt

The exact English translation depends on context.

In many everyday situations, English would naturally use the progressive:

• I am putting on the washed shirt

But German simply uses the present tense:

• Ich ziehe das gewaschene Hemd an.

This is a very common learner question.

• anziehen = put on
  the action of dressing

• tragen = wear
  the state of having something on

So:

• Ich ziehe das gewaschene Hemd an.
  = I am putting on the washed shirt.

• Ich trage das gewaschene Hemd.
  = I am wearing the washed shirt.

English separates these two ideas clearly, and German does too.

Yes. German word order is flexible, but the conjugated verb still has to stay in the second position in a main clause.

So you can say:

• Ich ziehe das gewaschene Hemd an.
• Das gewaschene Hemd ziehe ich an.

The second version emphasizes the washed shirt more.

What does not change is the separable-verb pattern:

• ziehe stays in second position
• an stays at the end

So even when the sentence starts with a different element, the verb rule still applies.