Відправник перевіряє адресу, а одержувач чекає посилку вдома.

Because they are the direct objects of the verbs, so they go into the accusative case.

• перевіряє що? → адресу
• чекає що? → посилку

For many feminine nouns ending in -а, the accusative singular changes to -у:

• адреса → адресу
• посилка → посилку

So this is a very common pattern in Ukrainian.

They are in the nominative singular, because they are the subjects of the two clauses:

• Відправник перевіряє адресу
• одержувач чекає посилку вдома

Both nouns name the person doing the action.

Both verbs are:

• 3rd person singular
• present tense
• imperfective aspect

That means:

• перевіряє = checks / is checking
• чекає = waits / is waiting

In Ukrainian, the present tense here can describe either:

• something happening right now, or
• something habitual/general

The exact meaning depends on context.

Here а is best understood as a contrastive and. It connects two ideas while slightly contrasting them:

• the sender is doing one thing,
• the recipient is doing another.

So in English it can sound like:

• and
• while
• whereas

It is not a strong but here. It is softer than that.

Because а joins two independent clauses:

• Відправник перевіряє адресу
• одержувач чекає посилку вдома

In Ukrainian, when two full clauses are connected by а, a comma is normally used.

Вдома is a very common adverb meaning at home.

So:

• чекає посилку вдома = waits for the package at home

You can sometimes hear expressions with дім / домі, but вдома is the most natural everyday way to say at home.

Yes. Чекати на + accusative is also very common:

• чекає посилку
• чекає на посилку

Both are used. The version in your sentence is perfectly normal and a little more compact.

No, Ukrainian word order is fairly flexible because the cases help show who is doing what.

The sentence you have is a very natural neutral order:

• Відправник перевіряє адресу, а одержувач чекає посилку вдома.

But other orders are possible if you want different emphasis, for example:

• Адресу перевіряє відправник, а посилку вдома чекає одержувач.

That sounds more marked and emphatic. For learners, the original version is the safest and most natural.

Because Ukrainian does not have articles like English the and a/an.

So:

• відправник can mean the sender or a sender
• адресу can mean the address or an address
• посилку can mean the package or a package

Context tells you which meaning is intended.

No. Another very common word is отримувач.

Both can mean recipient, and in many contexts they are very close in meaning. Depending on context, style, or region, one may sound more natural than the other.

In your sentence, одержувач is completely correct and natural.

Yes. Grammatically, both are masculine singular nouns.

That does not always mean the real person must be male; in some contexts masculine forms can be used in a general or role-based way. But grammatically, these particular forms are masculine.

You can often form feminine versions in Ukrainian too, depending on the word and the style of the situation.