Čemu služi ova mala tipka na daljinskom?

Because služiti in this meaning normally goes with the dative case.

• služiti čemu = to serve for something / to be used for something
• So the question word must also be in the dative:
  • što = what (base form)
  • čemu = to what / for what

So Čemu služi...? is literally something like To what does it serve?, which English usually expresses as What is it for?

Služi is the 3rd person singular present tense of služiti.

Here it means:

• serves
• is used for
• functions for

So in this sentence, služi is asking about the purpose of the button.

A useful pattern is:

• Ovo služi za rezanje. = This is used for cutting.
• Čemu to služi? = What is that for?

Because tipka is a feminine singular noun in the nominative, and both words before it agree with it:

• ova = this (feminine singular)
• mala = small (feminine singular)
• tipka = button/key

So all three words match in gender, number, and case.

If the noun were masculine, the forms would change:

• ovaj mali gumb = this small button

If it were neuter:

• ovo malo dugme = this small button (depending on regional vocabulary)

It is in the nominative singular because it is the grammatical subject of the sentence.

The structure is basically:

• Čemu služi = What is ... for?
• ova mala tipka = this small button
• na daljinskom = on the remote

So the thing that serves / is used is ova mala tipka, which makes it the subject.

Yes, in a way. This is a very common shortened everyday expression.

The full form would be:

• na daljinskom upravljaču = on the remote control

But in normal speech, people often shorten that to just:

• na daljinskom

Here daljinskom is standing in for the full phrase daljinskom upravljaču.

So the sentence is perfectly natural and idiomatic.

Here daljinskom is in the locative singular.

That is because na often takes the locative when it means on in the sense of location:

• na stolu = on the table
• na zidu = on the wall
• na daljinskom = on the remote

So this is not motion toward something; it is a location, which is why the locative is used.

Yes. That is also understandable and natural.

There is a slight difference in feel:

• Čemu služi...? is very compact and idiomatic
• Za što služi...? is a bit more explicit, closer to What is it used for?

Both are good Croatian. In everyday speech, Čemu služi...? is very common.

Both can often be translated as button, but they are not always used in exactly the same way.

Very roughly:

• tipka often refers to a button/key you press, especially on devices, keyboards, remotes, phones, etc.
• gumb can also mean button, but often more generally, and it also means a clothing button

Examples:

• tipka na tipkovnici = a key on a keyboard
• tipka na daljinskom = a button on a remote
• gumb na košulji = a button on a shirt

In this sentence, tipka sounds very natural for a remote-control button.

Because Croatian does not have articles like a and the.

English says:

• the remote
• the button

Croatian usually expresses definiteness through:

• context
• word order
• demonstratives such as ovaj / ova / ovo = this

Here, ova mala tipka already makes the button specific: this small button. And na daljinskom is understood from context as on the remote.

So Croatian does not need a separate word for the.

To some extent, yes. Croatian word order is more flexible than English word order.

The version you have is very natural:

• Čemu služi ova mala tipka na daljinskom?

The most important thing is that the question word čemu normally comes near the front, especially in a direct question.

You could also hear small variations depending on emphasis, but the given sentence is the most neutral and natural one for everyday speech.

The main sounds English speakers usually notice are:

• č in Čemu: like ch in church
• ž in služi: like the s in measure
• lj in daljinskom: a soft ly-type sound

A rough English-friendly pronunciation would be:

• Čemu ≈ CHEH-moo
• služi ≈ SLOO-zhi
• ova mala tipka ≈ OH-va MAH-la TEEP-ka
• na daljinskom ≈ na dal-YEEN-skom

That is only an approximation, but it is a useful starting point.