Şef bageti koro şefinin elinde.

Turkish often leaves out is/are in the present tense when the sentence is a noun or location sentence.

So:

Şef bageti koro şefinin elinde.

literally works like:

The conductor’s baton [is] in the choir conductor’s hand.

This is completely normal Turkish.
If you want a more formal or emphatic version, you can say:

Şef bageti koro şefinin elindedir.

The -dir adds the idea of is, but in everyday speech it is usually omitted.

Here, -i is not the accusative ending.

In şef bageti, the word bageti has the ending used in an indefinite noun compound. This is a very common Turkish structure.

Pattern:

Noun + Noun-(s)I

Examples:

• koro şefi = choir conductor
• çay bardağı = tea glass
• otobüs durağı = bus stop
• şef bageti = conductor’s baton

So:

• baget = baton
• bageti = baton, marked as the second part of the compound

This is one of the first things that feels unusual to English speakers, because English usually just puts two nouns together with no extra ending.

These mean slightly different things.

• şef bageti = conductor’s baton, baton used by a conductor, a baton identified by type/function
• şefin bageti = the baton of the conductor, belonging to a specific conductor

So şef bageti is a more general compound, while şefin bageti points to a specific possessor.

English often uses the same translation for both, so this difference can be easy to miss.

In your sentence, şef bageti names the object, and koro şefinin elinde tells you where it is.

koro şefinin means of the choir conductor or the choir conductor’s.

Breakdown:

• koro = choir
• şefi = conductor/leader of the choir
• şefinin = of the conductor / the conductor’s

More precisely:

• koro şefi = choir conductor
• koro şefinin = the choir conductor’s

The ending -nin / -ın / -in / -un / -ün is the genitive ending, which often corresponds to English ’s or of.

elinde breaks down like this:

• el = hand
• -i = his/her/its
• -n- = buffer consonant
• -de = in / at / on

So:

elinde = in his/her hand

Because the sentence already has koro şefinin, the full meaning becomes:

in the choir conductor’s hand

This is a very common Turkish pattern:

• evinde = in his/her house
• cebinde = in his/her pocket
• masasında = on his/her table

The -n- is a buffer consonant.

When a possessed noun takes another ending, Turkish often inserts -n- between them to make pronunciation smoother.

So:

• eli = his/her hand
• eli + de would be awkward
• Turkish uses elinde

This same thing happens in many words:

• evi → evinde
• arabası → arabasında
• kitabı → kitabında

So the -n- is not a separate meaning by itself here; it is just a connector.

Yes, by itself elinde normally means in his/her/its hand.

Turkish often leaves pronouns unstated when the context makes them clear.

Compare:

• elimde = in my hand
• elinde = in your hand / in his/her hand, depending on context
• elimizde = in our hand / in our possession
• ellerinde = in their hands

In your sentence, koro şefinin makes it clear whose hand it is, so elinde means in the choir conductor’s hand.

In Turkish, şef can mean several things depending on context, including:

• chief / boss
• chef
• conductor

In this sentence, because of baget and koro, the meaning is clearly conductor.

So şef bageti is a conductor’s baton, not something related to cooking.

For the same basic reason: both are the second word in an indefinite noun compound.

• koro şefi = choir conductor
• şef bageti = conductor’s baton

In both cases, Turkish marks the second noun with a possessive-looking ending, even though English does not.

So:

• şef + i in koro şefi
• baget + i in şef bageti

Then koro şefi can take another ending:

• koro şefinin = the choir conductor’s

This kind of stacking is very normal in Turkish, even though it can look complicated at first.

Yes. Turkish word order is flexible, although different orders can sound more natural in different contexts.

Your sentence:

Şef bageti koro şefinin elinde.

This is a natural way to say:

The conductor’s baton is in the choir conductor’s hand.

Another possible sentence is:

Koro şefinin elinde şef bageti var.

This is closer to:

There is a conductor’s baton in the choir conductor’s hand.

The meaning is similar, but the structure is different.
Your original sentence is a straightforward identification/location sentence, while the var version presents the baton more like something that exists in that location.

Yes, absolutely.

Şef bageti koro şefinin elindedir is correct.

The difference is mainly tone:

• elinde = more common in everyday speech
• elindedir = more formal, more written, sometimes more emphatic

Both mean the same thing here. In spoken Turkish, the shorter version without -dir is much more common.