Содержимое посылки уже проверили.

Because содержимое is the object, not the subject.

Russian is using an indefinite-personal construction here: the verb in the past plural means someone / they did the action, without naming who. So проверили does not agree with содержимое. It agrees with an implied, unnamed plural subject.

In other words, the sentence is structurally like:

• (Они) уже проверили содержимое посылки.
• They already checked the package contents.

But Russian normally leaves они out here.

Because Russian often leaves out pronouns when they are unnecessary, and this pattern very often omits the subject completely.

A past plural verb on its own can mean:

• they did it
• someone did it
• people did it

This is especially common when the doer is unknown, obvious, or unimportant.

So проверили by itself can mean they checked it without actually saying они.

It is in the accusative singular, because it is the direct object of проверили.

The verb проверить takes what was checked in the accusative:

• проверить документ
• проверить билет
• проверить содержимое

Here is the part that can confuse learners: for inanimate neuter nouns, the accusative looks exactly like the nominative. So содержимое does not change in form.

Because it depends on содержимое.

Russian says:

• содержимое чего? = the contents of what?

So посылки is genitive singular after содержимое.

This is a very common pattern in Russian with nouns showing content, part, amount, or relationship:

• часть книги = part of the book
• конец фильма = end of the film
• содержимое посылки = contents of the package

Because содержимое is a singular collective noun.

English often uses the plural contents, but Russian naturally uses one singular noun meaning what is inside. So grammatically it behaves like a normal singular noun:

• это содержимое
• всё содержимое
• содержимое посылки

So even if English thinks of it as plural, Russian does not have to.

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

That fits the idea of уже very well: the checking has already been done.

Compare:

• проверили = checked, finished checking
• проверяли = were checking / checked repeatedly / checked as a process

So in this sentence, проверили emphasizes the result: the action is done.

Уже means already.

It shows that the action happened before some expected or relevant point in time.

In this sentence, its position is neutral and natural:

• Содержимое посылки уже проверили.

But Russian word order is flexible, so уже can move depending on emphasis:

• Уже проверили содержимое посылки.
• Содержимое уже проверили.

The main meaning stays the same, but the information focus changes slightly.

Grammatically, it is active.

Russian literally says something like they already checked the contents of the package, with the subject left unnamed.

But in English, this kind of sentence is often translated more naturally as a passive:

• The contents of the package have already been checked.

So:

• Russian grammar: active, indefinite-personal
• Natural English translation: often passive

Almost, but not quite.

• Содержимое посылки уже проверили focuses on the completed action by unnamed people.
• Содержимое посылки уже проверено focuses on the resulting state: it is now in a checked condition.

So both can fit similar situations, but the nuance is slightly different.

Yes. Russian word order is flexible because case endings show the grammatical relationships.

This sentence is neutral and natural as:

• Содержимое посылки уже проверили.

But you could also say, for example:

• Уже проверили содержимое посылки.
• Содержимое уже проверили.
• Посылки содержимое уже проверили. — possible only in special contexts, but much less neutral

Changing the word order usually changes emphasis, not the basic meaning.