Stavi čaše na poslužavnik, molim te, da ih ne nosimo jednu po jednu.

Yes. Stavi is the 2nd person singular imperative of staviti (to put / to place).

So the speaker is talking to one person informally, like Put the glasses...

If the speaker were addressing:

• one person formally or more than one person, it would usually be stavite
• a close friend or family member, stavi is natural

So this sentence sounds like an informal request to one person.

Čaše is the accusative plural of čaša (glass), because it is the direct object of stavi.

You are putting the glasses, so glasses is the thing being acted on.

For many feminine nouns ending in -a, the nominative plural and accusative plural look the same:

• čaše = glasses as a subject
• čaše = glasses as a direct object

Compare the singular:

• čaša = nominative singular
• čašu = accusative singular

Because Croatian uses different cases after na depending on whether you mean:

• movement onto something → accusative
• location on something → locative

Here the meaning is Put the glasses onto the tray, so there is movement toward the tray:

• na poslužavnik = onto the tray

If you were just describing location, you would say:

• Čaše su na poslužavniku. = The glasses are on the tray.

So:

• na + accusative = direction / destination
• na + locative = position / location

Here molim te simply means please in an informal singular way.

Literally, it is related to I ask you or I beg you, but in everyday Croatian it often works just like a polite softener:

• Stavi čaše na poslužavnik, molim te. = Put the glasses on the tray, please.

If you were speaking formally or to more than one person, you would usually say:

• molim vas

Its position is flexible. You can hear it:

• in the middle of the sentence
• at the end
• sometimes near the beginning

Here da introduces a clause of purpose/result-like intention:

• da ih ne nosimo = so that we don’t carry them

In natural English, this often comes out as:

• so we don’t have to carry them
• so that we won’t be carrying them

This is a very common Croatian structure:

• Učini X, da se dogodi / ne dogodi Y.
• Do X, so that Y happens / doesn’t happen.

A more formal or more explicitly “so that we would not...” version might be something like:

• ...kako ih ne bismo nosili...

But da + present tense is extremely normal in everyday Croatian.

Because the speaker is including themselves in the situation.

The idea is:

• Put the glasses on the tray, so that we don’t carry them one by one.

So even though the command stavi is addressed to you, the consequence affects us:

• you put them on the tray
• then we won’t have to carry them individually

This is very natural in Croatian.

Ih is the unstressed pronoun meaning them. Here it refers back to čaše (the glasses).

So:

• da ih ne nosimo = so that we don’t carry them

Croatian usually uses this pronoun rather than leaving the object unspoken in this kind of clause.

Its position is also typical. Short pronouns like ih are clitics, which means they like to appear in a special unstressed position near the beginning of the clause:

• da ih ne nosimo

You do not normally say:

• da ne ih nosimo

Because this is the Croatian way to say one by one.

The phrase treats each item individually, so it uses the singular:

• jednu po jednu = one by one

It is feminine because čaša is feminine. So the form agrees with an understood singular item:

• one glass at a time

That is why you get:

• jednu po jednu for feminine nouns like čaša

Similar patterns:

• jedan po jedan for masculine
• jedno po jedno for neuter

Yes, but it is all very normal Croatian word order.

A few things to notice:

• Stavi čaše na poslužavnik: verb first is very common in commands.
• molim te can be inserted naturally as a politeness marker.
• da ih ne nosimo puts the short pronoun ih before the negated verb.
• jednu po jednu comes at the end, where it clearly explains how the glasses would otherwise be carried.

So the sentence sounds natural and idiomatic, not unusually formal or literary.