Anna ist erwachsen.

Because German uses sein (to be) with adjectives that describe a state or quality.

• Anna ist erwachsen. = Anna is an adult / Anna is grown up.

You would not use haben (to have) here, because erwachsen is not a thing Anna possesses. It describes what she is.

Erwachsen usually means adult, grown up, or fully mature.

So Anna ist erwachsen can mean:

• Anna is an adult
• Anna is grown up

Depending on context, it can refer:

• literally to age/status: she is no longer a child
• sometimes more loosely to maturity: she behaves like an adult

Yes. In Anna ist erwachsen, erwachsen is being used as an adjective after the verb sein.

This is called predicative use: the adjective comes after a linking verb like:

• sein = to be
• werden = to become
• bleiben = to remain

In this position, German adjectives do not take endings:

• Anna ist erwachsen.
• Die Kinder sind müde.
• Er bleibt ruhig.

Because it comes after ist and is used predicatively.

Compare:

• Anna ist erwachsen. → no ending
• eine erwachsene Anna → adjective before a noun, so it gets an ending
• eine erwachsene Frau → adjective before a noun, so it gets an ending

So the rule is:

• after sein/werden/bleiben → usually no adjective ending
• before a noun → adjective ending required

Yes, you can, but it means something slightly different in structure.

• Anna ist erwachsen. → Anna is grown up / adult.
  Here erwachsen is an adjective.

• Anna ist eine Erwachsene. → Anna is an adult.
  Here Erwachsene is being used like a noun.

Both are correct. The first is often more natural as a simple description. The second emphasizes her status as an adult person.

Because German main clauses usually put the conjugated verb in second position.

In Anna ist erwachsen:

1. Anna
2. ist
3. erwachsen

This is the normal word order for a simple statement.

You could also move another element to the first position in other sentences, but the verb would still stay second.

Yes. Erwachsen is historically related to wachsen, which means to grow.

• wachsen = to grow
• erwachsen = grown / grown up / adult

This helps explain why erwachsen can feel close to grown up in English.

A rough English-friendly guide is:

air-VAK-sen

More carefully:

• er sounds somewhat like air
• wach has the ch sound heard in German Bach, not like English ch in chair
• stress is on wach: er-WACH-sen

And the full sentence:

• Anna ist erwachsen.
• roughly: AHN-na ist air-VAK-sen

Not in this kind of sentence.

• Anna ist erwachsen.
• Die Kinder sind erwachsen.

The adjective stays erwachsen because it is used predicatively.

Only the verb changes:

• ist for singular
• sind for plural

Yes, sometimes.

Usually it means someone is literally an adult, but in context it can also suggest maturity:

• Sei erwachsen! = Be mature! / Act like an adult!

So the word can refer either to:

• actual adult status
• adult-like, mature behavior

In Anna ist erwachsen, the most basic reading is simply that Anna is an adult.

Because Anna is a proper name, and proper names are capitalized.

Erwachsen is not capitalized here because it is an adjective in the sentence:

• Anna ist erwachsen.

But if the word is used as a noun, it is capitalized:

• Anna ist eine Erwachsene.

This is a very important German rule:

• nouns are capitalized
• adjectives are not, unless they are functioning as nouns or starting the sentence

Yes. It is completely natural and grammatically complete.

It is a short, simple statement with:

• Anna = subject
• ist = verb
• erwachsen = predicate adjective

German often uses short sentences like this very naturally.