Haziran ve Temmuz arasında birkaç gün izin almalıymışım, yoksa çok yorulacağım.

It means between June and July.

The key word is arasında, which means between / among. In this sentence, it links the two months:

• Haziran = June
• Temmuz = July
• arasında = between

So the whole phrase is a time expression meaning in the period between June and July.

Because those mean different things.

• Haziran ve Temmuz arasında = between June and July
• Haziran'da ve Temmuz'da = in June and in July

So arasında creates a relationship between the two nouns, rather than saying something happens separately in each month.

Yes. Both are natural.

• Haziran ve Temmuz arasında
• Haziran ile Temmuz arasında

Both mean between June and July.
Using ve is a simple and. Using ile also links the two nouns and often sounds slightly more explicitly like between X and Y.

Because after numbers and quantifiers like birkaç, Turkish normally uses the singular form of the noun.

So:

• birkaç gün = a few days
• not birkaç günler

This is very common in Turkish:

• iki kitap = two books
• üç arkadaş = three friends
• birkaç soru = a few questions

Even though the meaning is plural in English, the Turkish noun usually stays singular.

İzin almak is a very common expression meaning to take leave, to get permission, or in this kind of context, to take time off.

Breakdown:

• izin = permission, leave
• almak = to take, to get

So birkaç gün izin almak means to take a few days off.

It is a fixed expression, so it is better to learn it as a chunk.

It can be broken down like this:

• al- = take
• -malı = should / must
• -y- = buffer consonant
• -mış = reported / inferred / apparently
• -ım = I

So almalıymışım means something like:

• apparently I should take
• it seems I should take
• I’ve been told I should take

The exact English translation depends on context, but the important point is that it is not as direct as plain almalıyım.

This is an important nuance.

• almalıyım = I should take / I must take
• almalıymışım = apparently I should take / it seems I should take / I’ve learned that I should take

So -mış adds a sense of:

• hearsay
• realization
• indirect knowledge
• a slightly less direct stance

For example:

• Almalıyım. = I personally know/think I should take it.
• Almalıymışım. = Apparently I should take it; that’s what I’ve realized / been told.

That y is a buffer consonant. Turkish often inserts y between vowels to make pronunciation smoother.

Without getting too technical, Turkish does this in many places:

• iyi + im → iyiyim
• almalı + im → almalıyım
• almalı + imişim → almalıymışım

So the y does not add meaning by itself. It is just there to connect sounds naturally.

Here yoksa means otherwise or or else.

So:

• ... izin almalıymışım, yoksa çok yorulacağım.
• ... I should apparently take some days off, otherwise I’ll get very tired.

It introduces a negative consequence if the first thing does not happen.

It comes from the verb yorulmak, which means to get tired or to become tired.

Breakdown:

• yorul- = get tired
• -acak / -ecek = future tense
• -ım = I

So:

• yorulacağım = I will get tired / I will become very tired

Notice that Turkish uses yorulmak, not yormak, because the speaker is the one becoming tired.

• yormak = to tire someone/something out
• yorulmak = to get tired

Both can make sense, but they are a little different.

• yorulacağım = I will get tired
• yorgun olacağım = I will be tired

So yorulacağım focuses on the process or result of becoming tired. It sounds very natural in a sentence about what will happen if the speaker does not take time off.

Because the speaker is talking about a likely future consequence.

• yoksa çok yorulacağım = otherwise I will get very tired

This is a prediction about what will happen later.

A different tense could slightly change the feel. For example, yorulurum could sound more general, like otherwise I get tired / I tend to get tired. But yorulacağım sounds more specific and concrete about the future.

No, Turkish word order is flexible, but the original order sounds natural and clear.

• Haziran ve Temmuz arasında birkaç gün izin almalıymışım, yoksa çok yorulacağım.

Putting Haziran ve Temmuz arasında at the beginning sets the time frame first, which is very common in Turkish.

You could move some parts around, but that would change emphasis. The original version is a good, natural way to say it.