Коворкинг открыт, и рабочее место готово.

In Russian, the verb to be is usually omitted in the present tense.

So:

• Коворкинг открыт = The coworking space is open
• рабочее место готово = the workspace is ready

Russian speakers normally do not say есть here. Using есть would sound unnatural in this kind of sentence.

Открыт is the short form of открытый.

Russian often uses short-form adjectives (and adjective-like forms) after the subject, especially when giving a state or condition:

• магазин открыт = the store is open
• дверь закрыта = the door is closed

So in this sentence, открыт means open as a current state.

By contrast, открытый is the full form, and it is more often used before a noun:

• открытый коворкинг = an open coworking space

So Коворкинг открыт is the natural predicative structure.

For the same reason: готово is the short form of готовый and is used as a predicate, meaning is ready.

Compare:

• готовый проект = a ready/finished project
• Проект готов = The project is ready

In your sentence:

• рабочее место готово = the workspace is ready

So готово is correct because it describes the state of the subject.

Because they agree with different nouns.

• коворкинг is masculine singular, so: открыт
• место is neuter singular, so: готово

Short-form adjectives agree in gender and number:

• masculine: готов
• feminine: готова
• neuter: готово
• plural: готовы

And similarly:

• masculine: открыт
• feminine: открыта
• neuter: открыто
• plural: открыты

The head noun is место, and место is a neuter noun in Russian.

Because of that, the adjective must also be neuter:

• рабочее место

Here:

• рабочее = neuter singular form of рабочий
• место = neuter singular noun

So the whole phrase is neuter singular, which is why the predicate is also neuter:

• рабочее место готово

Both subject phrases are in the nominative case:

• Коворкинг
• рабочее место

This is because they are the subjects of the two clauses:

• Коворкинг открыт
• рабочее место готово

The short forms открыт and готово do not show case the way full adjectives do. They mainly show gender and number.

Because и is joining two independent clauses:

• Коворкинг открыт
• рабочее место готово

Each part could stand on its own as a full sentence, so Russian normally uses a comma before и in this situation.

This is similar to English:

• The coworking space is open, and the workspace is ready.

Literally, it means working place or work place.

Depending on context, it can mean:

• a workspace
• a desk/workstation
• sometimes even a job position

In this sentence, because the context is коворкинг, it most naturally means a workspace or desk in the coworking space.

Yes. It is a borrowed word from English, but it is now commonly used in Russian.

It behaves like a normal masculine noun because it ends in a consonant:

• коворкинг
• коворкинга
• коворкингу, etc.

So in the sentence, коворкинг is masculine singular, which is why the predicate is открыт.

Yes, Russian word order is fairly flexible, but the original version is the most neutral:

• Коворкинг открыт, и рабочее место готово.

You could change the order for emphasis, but it may sound more marked or stylistic. For example:

• Рабочее место готово, и коворкинг открыт.

That changes the focus slightly by putting рабочее место first.

For a learner, the given order is the safest and most natural choice.

Not in normal Russian for this meaning.

Those are full forms:

• открытый
• готовое

Full forms usually describe nouns attributively, like:

• открытый коворкинг
• готовое рабочее место

But when you mean is open or is ready, Russian normally prefers the short forms:

• Коворкинг открыт
• рабочее место готово

So your original sentence uses the natural pattern.