Elif gelinceye kadar ben kütüphanede bekleyeceğim.

Because gelince and gelinceye kadar mean different things.

• gelince = when / once she comes
• gelinceye kadar = until she comes

So:

• Elif gelince = when Elif comes
• Elif gelinceye kadar = until Elif comes

The extra part -ye kadar changes the meaning from when to until.

That -ye is there because kadar normally follows the dative form.

You can see the same pattern with nouns:

• akşama kadar = until evening
• eve kadar = up to the house
• yarına kadar = until tomorrow

Here, the thing before kadar is not a noun but a verb-based form:

• gelince = when she comes
• gelinceye kadar = until she comes

So it helps to think of -ye kadar as a chunk meaning up to / until.

Turkish usually does not need subject pronouns if the subject is already clear.

In this sentence, Elif is the subject of gelinceye kadar, so Turkish does not need to add o.

So:

• Elif gelinceye kadar = until Elif comes
• not Elif o gelinceye kadar

That would be unnecessary.

No. Ben is optional here.

The verb bekleyeceğim already means I will wait, because the ending shows first person singular.

So both are possible:

• Elif gelinceye kadar ben kütüphanede bekleyeceğim.
• Elif gelinceye kadar kütüphanede bekleyeceğim.

Adding ben gives a little extra emphasis or contrast, something like:

• As for me, I’ll wait in the library
• I’ll be the one waiting in the library

Kütüphanede breaks down like this:

• kütüphane = library
• -de = in / at

So kütüphanede means in the library or at the library.

This is the locative ending, used for location.

More examples:

• evde = at home / in the house
• okulda = at school
• masada = on the table / at the table

It is built from:

• bekle- = wait
• -(y)ecek = future tense marker
• -im = I

So:

• bekle-
• bekleyecek
• bekleyeceğim = I will wait

A small sound change happens when endings are added, so you get -ceğ- in the final form.

You do not need to memorize every sound change immediately; the important point is:

• bekleyeceğim = I will wait

Because Turkish often uses a subordinate clause form instead of a future tense form in time expressions.

English says:

• until Elif comes
• not usually until Elif will come

Turkish works similarly here. It uses gelinceye kadar, not a future form like gelecek.

So even though the whole situation is in the future, the clause with until uses its own structure.

This is very common in Turkish:

• Sen gelince ararım. = I’ll call when you come.
• O dönene kadar bekleriz. = We’ll wait until he/she returns.

Yes. Elif gelene kadar is also correct and very common.

Both of these mean until Elif comes:

• Elif gelinceye kadar
• Elif gelene kadar

Many learners hear gelene kadar more often in everyday speech, but gelinceye kadar is also natural.

A useful rough distinction is:

• gelene kadar = very common everyday pattern
• gelinceye kadar = also correct, sometimes feels a bit fuller or more explicit

As a learner, you should understand both.

The verb is normally last, but Turkish word order is fairly flexible.

This sentence has a very natural order:

• Elif gelinceye kadar ben kütüphanede bekleyeceğim.

You could also say:

• Ben Elif gelinceye kadar kütüphanede bekleyeceğim.
• Elif gelinceye kadar kütüphanede bekleyeceğim.

These all work, but the emphasis changes a little.

A good rule for learners:

• keep the main verb at the end
• put the until-clause before the main clause if you want the safest, most natural order

Turkish does not have a word exactly like English the.

So kütüphanede can mean:

• in the library
• in a library

The exact English choice depends on context.

In this sentence, English naturally says the library, but Turkish just uses kütüphanede and lets context do the work.

Because Elif comes right before the subordinate verb form gelinceye kadar, so it is understood as the subject of that clause:

• Elif gelinceye kadar = until Elif comes

Then the main clause is:

• ben kütüphanede bekleyeceğim = I will wait in the library

So the sentence has two parts:

1. Elif gelinceye kadar
2. ben kütüphanede bekleyeceğim

That is a very common Turkish sentence pattern: a time clause first, then the main statement.