Недоимку и доначисление уже проверили.

Because both words are the direct objects of проверили (checked), so Russian puts them in the accusative case.

• недоимка → недоимку
  This is a feminine singular noun, and its accusative singular ends in -у.
• доначисление → доначисление
  This is a neuter singular inanimate noun, and in this pattern the accusative singular looks the same as the nominative singular.

So both are accusative, even though only one of them visibly changes.

They follow different noun patterns.

• Feminine nouns ending in -а often change in the accusative singular:
  недоимка → недоимку
• Neuter inanimate nouns ending in -е often have the same nominative and accusative singular:
  доначисление → доначисление

So the lack of visible change in доначисление is completely normal.

No. The verb is not plural because there are two objects.

In Russian, past-tense verbs agree with the subject, not the object. So проверили is plural because the implied subject is plural: they checked.

That means:

• проверили = they checked
• not it checked two things

The objects недоимку и доначисление do not control the verb form.

The subject is simply left unstated.

Russian often omits the subject when it is:

• obvious from context,
• unknown,
• unimportant,
• or intentionally vague.

So проверили can mean:

• they checked
• someone checked
• and in natural English often: it has already been checked / they have already been checked

This is very common in Russian, especially in official, business, or matter-of-fact statements.

Because проверили is perfective, so it presents the action as completed.

Here the meaning is roughly:

• they already checked it
• the check has been done

If you used проверяли, that would be imperfective, which would suggest something like:

• they were checking
• they used to check
• or a past action viewed as a process rather than a finished result

With уже, the perfective проверили sounds especially natural, because already often goes together with the idea of a completed action.

Уже means already.

It tells you that the checking happened before now or before some other reference point in the conversation.

So the sentence has the sense:

• The underpayment and the additional assessment have already been checked.

Russian word order is flexible, so уже can appear in different places, but here it naturally modifies the whole action проверили.

Russian word order is much freer than English word order because case endings show each word’s role.

Starting with Недоимку и доначисление puts those items first as the topic of the sentence: these are the things we are talking about. Then уже проверили gives the new information about them.

So the feeling is something like:

• As for the underpayment and the additional assessment, they’ve already checked them.

A different word order would still be possible, but the emphasis would shift.

Yes, very naturally.

Although the Russian sentence is formally active (they checked), English often prefers a passive when the doer is unknown or unimportant:

• The underpayment and the additional assessment have already been checked.

That is often the most natural English version in administrative or business contexts.

With stress marks, it is:

Недои́мку и доначисле́ние уже́ прове́рили.

Approximate pronunciation:

• недои́мку = ne-da-EEM-koo
• и = ee
• доначисле́ние = da-na-chees-LYE-ni-ye
• уже́ = oo-ZHE
• прове́рили = pra-VYE-ri-li

The sentence contains fairly formal, specialized vocabulary, so it sounds natural in accounting, tax, or bureaucratic contexts.