Wir prüfen die Verfügbarkeit eines barrierefreien Raums.

Because Verfügbarkeit is the direct object of prüfen, and prüfen takes the accusative case.

• die Verfügbarkeit = accusative singular
• die Verfügbarkeit also happens to look the same as nominative singular, because the noun is feminine

So:

• die Verfügbarkeit = nominative/accusative feminine singular
• der Verfügbarkeit = genitive/dative feminine singular

In this sentence, we are checking what? → the availability, so accusative is needed.

This phrase is in the genitive case, because it means the availability of an accessible room.

So:

• die Verfügbarkeit = availability
• eines barrierefreien Raums = of an accessible room

This is a very common German structure:
[noun] + genitive = [noun] of ...

Examples:

• die Farbe des Autos = the color of the car
• die Größe des Hauses = the size of the house
• die Verfügbarkeit eines Raums = the availability of a room

Eines is the genitive singular form of the indefinite article for a masculine or neuter noun.

Since Raum is masculine (der Raum), the genitive singular article is:

• nominative: ein Raum
• accusative: einen Raum
• dative: einem Raum
• genitive: eines Raums

So eines is used because the sentence needs genitive masculine singular.

Because adjective endings in German depend on:

1. the case
2. the gender
3. the article

Here we have:

• eines = genitive masculine singular
• Raums = masculine singular
• adjective after ein-word article in this form takes -en

So:

• eines barrierefreien Raums

This is a standard adjective ending pattern.

Compare:

• ein barrierefreier Raum = a barrier-free / accessible room
• einen barrierefreien Raum = an accessible room
• mit einem barrierefreien Raum = with an accessible room
• die Verfügbarkeit eines barrierefreien Raums = the availability of an accessible room

Because Raum is in the genitive singular, and masculine/neuter nouns often add -s or -es in the genitive.

So:

• nominative: der Raum
• accusative: den Raum
• dative: dem Raum
• genitive: des Raums

With the indefinite article:

• eines Raums

So the -s marks the genitive.

Literally, barrierefrei means barrier-free.

In real usage, it usually means:

• accessible
• wheelchair-accessible
• designed so that people with disabilities can use it more easily

So ein barrierefreier Raum is usually best translated as:

• an accessible room

Even though English often prefers accessible, German commonly uses barrierefrei.

In this sentence, prüfen is best understood as to check or to verify.

Depending on context, prüfen can mean:

• check
• examine
• inspect
• test

Here, Wir prüfen die Verfügbarkeit ... means something like:

• We are checking the availability ...
• We will verify the availability ...

It does not mean physically testing the room itself. It means checking whether such a room is available.

German often uses the genitive after abstract nouns like Verfügbarkeit.

So:

• die Verfügbarkeit eines barrierefreien Raums = the availability of an accessible room

A von phrase is sometimes possible in German, but here the genitive is more natural and more standard.

Using von would sound less elegant:

• die Verfügbarkeit von einem barrierefreien Raum

That is understandable, but native speakers would usually prefer:

• die Verfügbarkeit eines barrierefreien Raums

So for formal or standard German, the genitive is the better choice here.

The structure is:

• Wir = subject
• prüfen = verb
• die Verfügbarkeit = direct object
• eines barrierefreien Raums = genitive phrase modifying Verfügbarkeit

So literally:

• We check the availability of an accessible room.

German main clauses normally put the conjugated verb in the second position, and that is exactly what happens here:

• Wir | prüfen | ...

Because they are nouns, and in German all nouns are capitalized.

Here:

• Verfügbarkeit = noun
• Raum / Raums = noun

But:

• prüfen is a verb, so it is not capitalized
• barrierefreien is an adjective, so it is not capitalized

This is one of the most noticeable spelling rules in German.

The base adjective is verfügbar, which means available.

From that, German forms the noun:

• die Verfügbarkeit = availability

So:

• verfügbar = available
• die Verfügbarkeit = availability

This is similar to English:

• available → availability

It is a useful pattern to notice, because German often forms abstract nouns this way.

Yes. That is a very natural alternative.

It means:

• We are checking whether an accessible room is available.

The difference is mainly style:

• Wir prüfen die Verfügbarkeit eines barrierefreien Raums.
  More compact, formal, noun-heavy

• Wir prüfen, ob ein barrierefreier Raum verfügbar ist.
  More verbal, often a bit more conversational and direct

Both are correct.

The plural of Raum is Räume.

So:

• singular: ein barrierefreier Raum = an accessible room
• plural: barrierefreie Räume = accessible rooms

If you wanted to say the availability of accessible rooms, you could say:

• die Verfügbarkeit barrierefreier Räume

Notice that in the plural, there is no article here, and the adjective ending changes:

• barrierefreier Räume

That is because the grammar is different from the singular phrase eines barrierefreien Raums.