Pronájem místnosti je drahý.

Because after the noun pronájem (rental / renting), Czech normally puts the thing being rented in the genitive case.

• místnost = a room
• místnosti = of a room

So pronájem místnosti literally means the rental of a room.

This is very common in Czech with action nouns:

• prodej domu = the sale of a house
• koupě auta = the purchase of a car
• pronájem bytu = the rental of an apartment

Because drahý describes pronájem, not místnosti.

The main subject of the sentence is pronájem, and pronájem is a masculine noun. So the adjective must agree with pronájem:

• pronájem → masculine
• therefore drahý → masculine singular

If the sentence were about the room itself, then you would use a feminine adjective:

• Místnost je drahá. = The room is expensive.

But here the meaning is:

• The rental of a room is expensive not
• The room is expensive

It is in the nominative singular, because it is the subject of the sentence.

You can think of the structure like this:

• Pronájem místnosti = subject
• je = is
• drahý = expensive

So the sentence is built around pronájem as the thing being talked about.

Je is the 3rd person singular form of být (to be), meaning is.

So:

• je = is

In this sentence it links the subject and the description:

• Pronájem místnosti = the rental of a room
• je drahý = is expensive

Czech often uses být in the same kind of way English uses to be.

Grammatically, yes, that is close to the structure. A more natural English translation would usually be:

• Renting a room is expensive
• The rental of a room is expensive
• Room rental is expensive

So the Czech sentence is very compact, but its structure is slightly closer to the rental of a room than to a room is expensive.

Here it is singular: of a room.

The form místnosti can belong to more than one case/number pattern in Czech, which is normal. But in this sentence, because it depends on pronájem, the intended reading is genitive singular.

So:

• místnost = room
• místnosti = of a room

Yes, that is possible, but nájem and pronájem are not always exactly the same in usage.

Very roughly:

• nájem often focuses on rent / tenancy / lease
• pronájem often focuses on rental / renting out / lease arrangement

In many everyday contexts, both may be understandable, but pronájem místnosti is a very natural phrase for room rental.

A learner should mainly remember:

• both are real words
• the exact nuance depends on context

Because after být (to be), Czech uses an adjective to describe the subject.

So:

• je drahý = is expensive

An adverb would not fit here, because the sentence is not describing how something happens; it is describing what the subject is like.

Compare:

• Pronájem je drahý. = The rental is expensive.
• Platí draze. = He/she pays expensively / at a high cost.

In your sentence, you need the adjective.

Yes. Czech word order is more flexible than English word order, although the neutral version here is:

• Pronájem místnosti je drahý.

You might also hear different word orders for emphasis, for example:

• Drahý je pronájem místnosti.
• Pronájem místnosti drahý je. (less neutral, more marked)

But for a learner, the standard and safest version is the original one.

A simple learner-friendly guide:

• Pronájem ≈ pro-NAA-yem
• místnosti ≈ MEEST-nos-ti
• je ≈ ye
• drahý ≈ DRA-hee

A few useful points:

• Czech stress is usually on the first syllable of each word.
• á, í, ý are long vowels.
• j is pronounced like English y in yes.

So the rhythm is roughly:

PRO-ná-jem MÍST-nos-ti ye DRA-hý

Yes, it is natural and grammatical.

It sounds like something you could see in a discussion about housing, prices, or renting. It is a straightforward neutral sentence meaning that renting a room costs a lot.

It may sound slightly formal or general because pronájem místnosti is a noun phrase. In everyday speech, people might also use other expressions depending on context, but this sentence itself is perfectly normal.