נורת האזהרה אדומה.

נורת is the construct form of נורה. Hebrew often links two nouns by putting the first noun into a special form called the construct state.

So:

נורה = a light / bulb / lamp
נורת האזהרה = literally the light of the warning, naturally the warning light

In this pattern, נורה changes to נורת.

In a construct chain, the first noun usually does not take ה־. Instead, definiteness is marked on the second noun, and that makes the whole phrase definite.

So:

נורת אזהרה = a warning light
נורת האזהרה = the warning light

הנורת האזהרה is not the standard form.

Yes, in a sense. The construct chain often corresponds to English of.

So נורת האזהרה can be understood literally as the light of the warning. But in natural English we usually say the warning light rather than the light of the warning.

Because the noun being described is נורה, and נורה is feminine singular.

Hebrew adjectives must agree with the noun in gender and number, so:

אדום = masculine singular
אדומה = feminine singular

Since נורת האזהרה is headed by נורה, the adjective must be אדומה.

It agrees with נורה, the main noun in the phrase.

In נורת האזהרה, the head noun is נורה and האזהרה just tells us what kind of light it is. So the adjective matches נורה, not the second noun.

In present-tense Hebrew, the verb to be is usually omitted.

So:

נורת האזהרה אדומה
literally looks like
the warning light red

but it means
The warning light is red.

In other tenses, Hebrew does use a form of to be:

נורת האזהרה הייתה אדומה = The warning light was red
נורת האזהרה תהיה אדומה = The warning light will be red

In Hebrew, adjectives usually come after the noun they describe.

When the noun is part of a construct chain, the adjective normally comes after the entire chain, not in the middle of it.

So:

נורת האזהרה אדומה

not something like putting אדומה between the two nouns.

The dictionary form is usually the masculine singular form: אדום.

Its common forms are:

אדום = masculine singular
אדומה = feminine singular
אדומים = masculine plural
אדומות = feminine plural

Yes. הנורה אדומה means the light / bulb is red.

That is a simpler noun + adjective sentence.

But נורת האזהרה אדומה is more specific: it means the warning light is red. The construct form נורת is what creates that more specific noun phrase.

No. You may also hear נורית אזהרה, especially for an indicator light on a dashboard or a device.

Both are natural, but נורת האזהרה is completely standard and correct.

It would be:

נורות האזהרה אדומות

That means The warning lights are red.

Notice that both the noun and the adjective change to feminine plural:

נורות = lights
אדומות = red, feminine plural