هذه امرأة.

Because the sentence means This is a woman, not this woman.

• هذه امرأة = This is a woman
• هذه المرأة = this woman or this is the woman

The absence of ال makes امرأة indefinite: a woman, not the woman.

The key is that امرأة is indefinite. In Arabic, a demonstrative followed by an indefinite noun often forms a complete sentence of identification:

• هذه امرأة = This is a woman

If you want the noun phrase this woman, Arabic normally uses a definite noun:

• هذه المرأة = this woman

This is a nominal sentence.

• هذه is the topic / subject-like part
• امرأة is the predicate, telling you what this is

So the structure is essentially:

• this
  • a woman

which in natural English becomes This is a woman.

It would be nominative, because it is the predicate of a nominal sentence:

• امرأةٌ

In normal Arabic writing, short vowels and case endings are usually omitted, so you normally just see امرأة.

On its own, it is commonly pronounced imra’ah in pause.

If fully inflected in careful reading, it can be imra’atun. This word has a somewhat irregular spelling and pronunciation, so it is one of those common words that learners usually just memorize.

That is tā’ marbūṭa, a very common ending on feminine nouns.

In pronunciation:

• in pause, it usually sounds like -ah
• when grammatical endings are pronounced, it can sound like -t

So:

• امرأة in pause: imra’ah
• امرأةٌ in full pronunciation: imra’atun

Yes. In Modern Standard Arabic, demonstratives like هذا and هذه normally come before the noun.

For example:

• هذه المرأة = this woman

So the word order here is normal Arabic word order. The reason it means This is a woman is not the position of the words, but the fact that امرأة is indefinite.