المعطف الواقي من المطر في حقيبة الظهر.

Yes. In Arabic, the present-tense verb to be is usually not stated.

So المعطف الواقي من المطر في حقيبة الظهر is a normal nominal sentence:

• المعطف الواقي من المطر = the raincoat
• في حقيبة الظهر = in the backpack

So the sentence means The raincoat is in the backpack, even though Arabic does not say a separate word for is here.

A rough breakdown is:

• المعطف = the coat
• الواقي = protective / protecting
• من المطر = from the rain

So literally it is something like the coat that protects from the rain.
In natural English, that is simply the raincoat.

Because الواقي is describing المعطف, so it works like an adjective here, and Arabic adjectives usually agree with the noun they describe.

Since المعطف is:

• definite: it has الـ
• masculine
• singular

the adjective must match it, so we get:

• المعطف الواقي

If the noun were indefinite, you would normally make the adjective indefinite too:

• معطف واقٍ من المطر = a raincoat / a coat protective against rain

It is originally an active participle from the verb وقى = to protect / shield.

But in sentences like this, it behaves very much like an adjective and is often best understood as:

• protective
• protecting
• shielding

So in practice, for a learner, it is fine to think of الواقي here as an adjective modifying المعطف.

من المطر means from the rain, and it completes the meaning of الواقي.

Without it, الواقي just means protective.
With من المطر, it becomes protective from the rain or rain-protective.

So:

• معطف واقٍ = a protective coat
• معطف واقٍ من المطر = a coat that protects from rain / a raincoat

Arabic often uses الـ with things that English treats more generally, especially natural phenomena and generic ideas.

So المطر can mean rain in a general sense, not only the specific rain.

That means:

• من المطر is very natural for from the rain
• من مطر is possible in some contexts, but it sounds less like the standard expression here

So in this sentence, من المطر is the normal, idiomatic choice.

Because حقيبة الظهر is an iḍāfa construction, often called a construct phrase.

It literally means:

• حقيبة = bag
• الظهر = the back

So حقيبة الظهر = bag of the back, which is the Arabic way to say backpack.

In an iḍāfa:

• the first noun usually does not take الـ
• the second noun can take الـ
• the whole phrase can still be definite

So:

• حقيبة ظهر = a backpack
• حقيبة الظهر = the backpack

That is why حقيبة has no الـ, even though the whole phrase is definite.

Because حقيبة is followed by another noun in an iḍāfa construction.

The final ة (tāʾ marbūṭa) is often pronounced like -a / -ah when stopping on the word by itself:

• حقيبة = ḥaqība

But when the word is grammatically connected to what follows, especially in iḍāfa, the ة is pronounced as t:

• حقيبة الظهر = ḥaqībat aẓ-ẓahr

This is very common in Arabic.

Because ظ is a sun letter.

With sun letters, the ل of الـ is not pronounced, and the next consonant is doubled. So:

• الظهر is written with الـ
• but pronounced aẓ-ẓahr

By contrast, in المطر, the م is a moon letter, so the ل is pronounced:

• المطر = al-maṭar

So the pronunciation of the whole phrase is approximately:

• al-miʿṭaf al-wāqī min al-maṭar fī ḥaqībat aẓ-ẓahr

Here is the basic structure:

• المعطف = the main noun, the topic/subject of the sentence
• الواقي = adjective describing المعطف
• من المطر = prepositional phrase linked to الواقي
• في حقيبة الظهر = the predicate, telling where the raincoat is

So the sentence is built like this:

• [subject noun phrase] + [location phrase]

More specifically:

• المعطف الواقي من المطر = the raincoat
• في حقيبة الظهر = is in the backpack

Also:

• المطر is after من, so it is genitive
• حقيبة is after في, so it is genitive
• الظهر is genitive as the second noun in the iḍāfa

Yes. That would be the indefinite version.

Compare:

• معطف واقٍ من المطر = a raincoat
• المعطف الواقي من المطر = the raincoat

So the sentence you were given is definite: it refers to a specific raincoat, not just any raincoat.

Yes, Arabic allows that.

You could say:

• في حقيبة الظهر المعطف الواقي من المطر

That puts more emphasis on the location, something like:

• In the backpack is the raincoat

But the original order is the more neutral one for a beginner to understand:

• المعطف الواقي من المطر في حقيبة الظهر

Both are possible, but the original feels more straightforward and basic.

It is a very understandable and correct descriptive phrase, but it is somewhat transparent in structure:

• coat + protective + from rain

So it feels a bit like the coat that protects from rain.

That is perfectly good Arabic, and very useful for learners because the meaning is easy to see. In real usage, Arabic may also use other expressions for raincoat, but this sentence is grammatically clear and natural enough for learning how adjectives and prepositional phrases work together.