Kurier doręczy sześć przesyłek.

Kurier is in the nominative singular. It is the subject of the sentence, so the verb agrees with it.

• kurier = subject
• doręczy = verb
• sześć przesyłek = direct object

Because kurier is singular, the verb is singular too.

Because doręczyć is a perfective verb.

In Polish, perfective verbs do not normally describe an ongoing present action. Their present-looking forms usually refer to a completed action in the future.

So:

• doręczy = will deliver
• doręcza = is delivering / delivers (imperfective)

The imperfective partner is doręczać.

Because the verb agrees with the subject, not with the object.

Here the subject is kurier = courier, which is singular.
The phrase sześć przesyłek is only the object, so it does not control the verb form.

That is why you get:

• Kurier doręczy... = singular verb

not a plural verb.

After the numerals five and higher, the counted noun normally goes into the genitive plural.

So:

• przesyłka = one parcel/shipment
• dwie / trzy / cztery przesyłki = two / three / four parcels
• pięć / sześć przesyłek = five / six parcels

So przesyłek is the correct form after sześć.

Przesyłka is a fairly broad word. It can mean:

• parcel
• package
• shipment
• consignment

In a sentence with kurier, the most natural English choices are often parcel or package, but the exact translation depends on context.

Because Polish has no articles.

So kurier can mean either:

• a courier
• the courier

and przesyłek can mean:

• parcels
• the parcels

Context tells you which meaning is intended.

Yes. Polish word order is quite flexible because case endings show what each word is doing.

The most neutral order here is:

• Kurier doręczy sześć przesyłek.

But other orders are possible for emphasis, for example:

• Sześć przesyłek doręczy kurier.
  This puts more focus on six parcels or contrasts the courier with someone else.

So the original sentence is the most neutral, but not the only possible order.

Yes, and that would also be natural.

The difference is mainly one of nuance:

• doręczyć often emphasizes handing something over to the recipient, and can sound a bit more official or postal/courier-like
• dostarczyć is broader and more general: to deliver / bring / supply

In many everyday contexts, both are possible.

A rough pronunciation is:

Kurier doręczy sześć przesyłek
/ˈku.rjɛr dɔˈrɛn.t͡ʂɨ ʂɛɕt͡ɕ pʂɛˈsɨ.wɛk/

A few useful sound notes:

• sz sounds like English sh
• cz sounds roughly like English ch
• ł sounds like English w
• ę before cz in doręczy is pronounced roughly like en
• prz at the start of przesyłek sounds roughly like psh

The trickiest word for many learners is sześć.

The normal negative version is:

Kurier nie doręczy sześciu przesyłek.

A useful thing to notice is that under negation, the object phrase changes:

• sześć przesyłek → sześciu przesyłek

This is a more advanced grammar point, but it is the standard pattern here.