Dostępność i kolejkę oczekujących sprawdzę w katalogu.

Because sprawdzę is the 1st person singular form of the perfective verb sprawdzić.

In Polish, perfective verbs do not have a present tense with present meaning. Their present-looking forms are used for the future. So:

• sprawdzam = I am checking / I check (imperfective, present)
• sprawdzę = I will check (perfective, future)

So sprawdzę means a completed future action: I’ll check.

Because kolejkę is the accusative singular of kolejka.

It is a direct object of sprawdzę, and direct objects often take the accusative in Polish.

• nominative: kolejka
• accusative: kolejkę

So in this sentence, kolejkę oczekujących is one of the things being checked.

Because dostępność belongs to a different declension pattern from kolejka.

It is also in the accusative here, but for many feminine nouns ending in -ość, the nominative and accusative singular have the same form.

So:

• nominative: dostępność
• accusative: dostępność

That is why the sentence has:

• dostępność
• kolejkę

Even though both are objects, they look different because they decline differently.

They are actually in the same case: both are accusative singular.

They just have different endings because they are different kinds of nouns:

• dostępność → accusative looks the same as nominative
• kolejka → accusative changes to kolejkę

So grammatically they match: both are direct objects of sprawdzę.

Oczekujących comes from oczekujący, which means waiting.

Here it functions like of those who are waiting or waiting people. In the phrase:

• kolejkę oczekujących

it means something like:

• the queue/list of people waiting

Grammatically, oczekujących is in the genitive plural, because it depends on kolejka:
a queue of whom? → of the waiting people

So this is a compressed Polish way of saying the waiting list / the queue of people waiting.

It can be natural, but it depends on context.

• kolejka oczekujących suggests a queue or line of people waiting
• lista oczekujących is often the more direct equivalent of waiting list

So if the situation is formal or administrative, many speakers might prefer:

• listę oczekujących

But kolejkę oczekujących is understandable and can sound natural if the idea is more like a queue of people waiting for something.

Because w means in here, and when it expresses location, it takes the locative case.

• dictionary form: katalog
• locative singular: katalogu

So:

• w katalogu = in the catalog / in the directory

If there were motion into something, Polish might use a different structure, but here it is static location: the checking happens in the catalog.

Because Polish verb endings usually already show the subject.

The ending -ę in sprawdzę tells you it means I will check.

So ja is usually unnecessary unless you want emphasis or contrast, for example:

• Ja sprawdzę, a ty zadzwoń.
  I’ll check, and you call.

In a neutral sentence, leaving out ja is normal.

Yes, it is natural. Polish word order is much more flexible than English word order.

This sentence puts the focus first on what will be checked:

• Dostępność i kolejkę oczekujących → the things being checked
• sprawdzę → the action
• w katalogu → where

A more neutral or equally natural version could also be:

• Sprawdzę dostępność i kolejkę oczekujących w katalogu.

The original version gives a bit more emphasis to the objects.

That depends on context. Katalog can mean things like:

• a catalog
• a directory
• a listing
• in some contexts, a library catalog

Because the sentence mentions availability and a waiting list/queue, it could easily refer to something like a library system, booking system, or database.

So w katalogu should be understood according to the situation, not as one fixed English word in every context.