Bekleme odası dar olsa da insanlar sakince sıralarını bekliyor.

It is a noun phrase meaning waiting room.

It is built from:

• beklemek = to wait
• bekleme = waiting
• oda = room

So bekleme odası is literally something like room for waiting or room of waiting, which English naturally says as waiting room.

Because Turkish often forms this kind of compound with the pattern:

Noun + Noun-(s)I

So:

• bekleme odası = waiting room
• çay bardağı = tea glass
• otobüs durağı = bus stop

The -sı here is part of the compound structure. It does not mean his/her room in this sentence.

Dar olsa da means although it is narrow or even though it is narrow.

In a simple statement, Turkish can say:

• Bekleme odası dar. = The waiting room is narrow.

But when Turkish wants to make a concessive structure like although..., it commonly uses olmak:

• dar olsa da = although it is narrow

Here:

• olsa comes from olmak with the conditional suffix -sa/-se
• da adds the concessive sense: even though / although

No. They are different things.

• -da/-de as a suffix is the locative ending:
  odada = in the room
• da in olsa da is a separate word and is part of the pattern meaning although / even though

You can tell because it is written separately:

• olsa da = separate particle
• odada = suffix attached to the noun

Yes, you could say:

Bekleme odası dar ama insanlar sakince sıralarını bekliyor.

That means roughly the same thing: The waiting room is narrow, but people are calmly waiting their turn.

The difference is mainly in style:

• ama = but, a straightforward connector
• olsa da = although / even though, a more tightly linked concessive structure

So both are natural, but olsa da sounds a bit closer to although even though.

Sakince means calmly or quietly.

It comes from:

• sakin = calm, quiet
• -ce/-ca = a suffix that often makes an adverb meaning in a ... way

So:

• sakin = calm
• sakince = calmly

Here sıralarını means their turn or their turns.

The noun sıra can mean things like:

• turn
• order
• place in line
• sometimes queue/row, depending on context

In this sentence, the natural meaning is their turn.

The very common expression is:

• sırasını beklemek = to wait one’s turn

Because the subject is plural, Turkish uses:

• sıralarını beklemek = to wait their turns

It can be analyzed like this:

• sıra = turn / order
• -lar = plural
• -ı = third-person possessive
• -nı = accusative after a possessive ending

So sıralarını is something like: their turns as a direct object.

In practice, you do not need to translate each piece separately every time. The important thing is that sıralarını bekliyor means are waiting their turn / their turns.

Because beklemek is a transitive verb here, so it takes a direct object.

Turkish often uses the accusative when the object is:

• specific
• definite
• or possessed

Since sıralarını means their turn(s), it is a possessed and specific object, so the accusative is expected.

Also, after a possessive ending, the accusative appears as:

• -nı / -ni / -nu / -nü

That is why you see sıralarını, not just sıraları.

Because in Turkish, when the plural subject is already stated, the verb often appears without the plural ending.

So both of these can be correct:

• İnsanlar bekliyor.
• İnsanlar bekliyorlar.

The version without -lar on the verb is very common and natural when the subject is already explicit.

A rough way to think of it is:

• insanlar bekliyor = the plural is already clear from insanlar
• bekliyorlar is especially useful when the subject is omitted and the verb itself has to show they

Bekliyor is in the present continuous/progressive.

Structure:

• bekle- = verb stem
• -iyor = present continuous

So:

• bekliyor = is waiting / are waiting

In this sentence, it describes an action happening at that moment: people are calmly waiting their turn.

Yes. It is very natural Turkish word order.

The sentence is basically:

• Bekleme odası dar olsa da = concessive clause: Although the waiting room is narrow
• insanlar = subject: people
• sakince = adverb: calmly
• sıralarını = object: their turn(s)
• bekliyor = verb: are waiting

Turkish often puts the verb at the end, and objects usually come before the verb. So insanlar sakince sıralarını bekliyor is a very normal structure.