В переговорной хорошая звукоизоляция.

Russian usually omits the present-tense form of to be.

So В переговорной хорошая звукоизоляция is a normal complete sentence even though there is no word for is. Russian often does this in sentences like:

• Комната большая.
• Кофе горячий.
• Здесь тихо.

In the past or future, a form of быть does appear:

• В переговорной была хорошая звукоизоляция.
• В переговорной будет хорошая звукоизоляция.

Because в is expressing location here, and after в for location Russian uses the prepositional case.

The base form is переговорная.
After в meaning in, it becomes в переговорной.

Compare:

• в переговорной = in the meeting room → location, so prepositional
• в переговорную = into the meeting room → motion/direction, so accusative

This location vs. motion contrast is very important in Russian.

Переговорная is a very common example of an adjective being used as a noun.

It originally comes from something like:

• переговорная комната = meeting room / negotiation room

Then комната is left out, and переговорная stays behind and functions like a noun by itself.

Because the hidden word комната is feminine, переговорная behaves as a feminine word. That is why its forms look like adjective endings:

• переговорная
• переговорной
• переговорную

So although it means a room, grammatically it still looks adjective-like.

Because хорошая is an adjective, and it describes the noun звукоизоляция.

• хорошая звукоизоляция = good soundproofing

The noun звукоизоляция is feminine singular, so the adjective must match it:

• хорошая = feminine singular
• хороший = masculine singular
• хорошее = neuter singular
• хорошие = plural

Хорошо is usually an adverb, so it would be used in a different structure, for example:

• Комната хорошо звукоизолирована. = The room is well soundproofed.

But in your sentence, the speaker is describing the noun звукоизоляция, so хорошая is the correct form.

It is in the nominative singular.

In this sentence, звукоизоляция is the thing being talked about, and хорошая agrees with it. The phrase в переговорной only gives the location.

You can think of the structure as:

• В переговорной = location
• хорошая звукоизоляция = the main noun phrase

So the sentence is built around the idea good soundproofing [is] in the meeting room.

No. Russian word order is flexible, and changing it changes the emphasis more than the basic meaning.

Your sentence:

• В переговорной хорошая звукоизоляция.

This sounds natural and neutral if you first want to set the scene: as for the meeting room, the soundproofing is good.

Other possible versions:

• Звукоизоляция в переговорной хорошая.
  Focuses more on the soundproofing.

• В переговорной звукоизоляция хорошая.
  More conversational; the adjective at the end can sound slightly more contrastive or emphatic.

So the original order is very natural, but not the only possible one.

Yes, that is grammatically possible, but it is not the most neutral version.

Adding есть puts more emphasis on existence / presence:

• В переговорной есть хорошая звукоизоляция.
  = There is good soundproofing in the meeting room.

Without есть, the sentence sounds more like a simple description of a feature:

• В переговорной хорошая звукоизоляция.
  = The meeting room has good soundproofing / The soundproofing in the meeting room is good.

In modern Russian, learners often overuse есть because English uses there is so much. In sentences like this, leaving it out is usually more natural.

Russian has no articles, so the sentence itself does not explicitly say the or a.

Context decides.

Depending on the situation, в переговорной could mean:

• in the meeting room
• in a meeting room

In real life, it often refers to a specific known room, because speaker and listener usually know which meeting room is meant.

If Russian wants to be more specific, it can use other words, for example:

• в этой переговорной = in this meeting room
• в одной переговорной = in one / a certain meeting room

So the lack of an article is normal and not a problem in Russian.

Not necessarily.

Звукоизоляция means sound insulation / soundproofing as a property or feature.
So хорошая звукоизоляция means the room blocks sound well, but it does not automatically mean perfectly soundproof.

If you want a different wording, Russian could also say something like:

• Переговорная хорошо звукоизолирована.
  = The meeting room is well soundproofed.

So your sentence is natural, but it describes the quality of the sound insulation rather than making an absolute claim that no sound gets through.