Kayıt ücreti ile geçerlilik süresini birlikte kontrol ettik.

Here ile is functioning more like and, linking kayıt ücreti and geçerlilik süresini.

So the sense is:

• the registration fee and the validity period

In other contexts, ile can also mean with, so it is a very common Turkish word with both uses. In this sentence, the coordinated meaning and is the natural reading.

Yes. İle can be written separately, as in your sentence, or attached as a clitic-like ending:

• kayıt ücreti ile
• kayıt ücretiyle

Both are correct. When attached, it usually becomes:

• -la / -le
• or -yla / -yle after a vowel

Since ücreti ends in a vowel, attached ile becomes -yle:

• ücretiyle

Because this is a very common Turkish noun-compound pattern.

• kayıt = registration
• ücret = fee
• kayıt ücreti = registration fee

In this pattern, the second noun usually takes the ending -(s)I:

• ücret → ücreti

This ending may look possessive at first, and historically it is related to that, but in modern Turkish it is simply the normal way many noun compounds are formed.

So:

• kayıt ücreti = registration fee
• not kayıt ücret

Because geçerlilik süresi is the base noun compound, and here it is being used as a definite direct object.

Step by step:

• geçerlilik = validity
• süre = duration / period
• geçerlilik süresi = validity period

Then Turkish adds the accusative ending because this object is specific:

• süresi → süresini

So:

• geçerlilik süresi = validity period
• geçerlilik süresini = the validity period, as the direct object

Because the word already contains the compound ending -si:

• süre → süresi

When a noun like that takes the accusative, Turkish uses a buffer n before the accusative ending:

• süresi + ni → süresini

So the n is not random. It appears because another suffix is already there.

This same thing happens often after possessive-type endings and noun-compound endings.

A learner often expects both nouns to be marked separately, but Turkish does not always do that with coordinated nouns.

Here the whole phrase:

• kayıt ücreti ile geçerlilik süresini

works as one object phrase, and the accusative marking appears on the last item.

So the sentence can still mean:

• We checked the registration fee and the validity period together.

Also, note that ücreti already ends in -i, but that -i is part of the noun compound kayıt ücreti, not the accusative marker.

Birlikte means together.

In this sentence it adds the idea that the two things were checked together / jointly / as a pair, rather than separately.

Without birlikte, the sentence would still be grammatical:

• Kayıt ücreti ile geçerlilik süresini kontrol ettik.

Adding birlikte makes the action feel more explicitly combined.

Because kontrol etmek is a very common Turkish compound verb.

• kontrol = check, control, inspection
• etmek = to do / to make

Together:

• kontrol etmek = to check

Turkish uses many expressions like this, especially with nouns of foreign origin:

• yardım etmek = to help
• telefon etmek = to phone
• kontrol etmek = to check

So kontrol ettik means we checked.

Ettik is:

• past tense
• 1st person plural

So it means:

• we did
• and here, as part of kontrol ettik, we checked

Breakdown:

• et- = do
• -ti / -dı type ending = past tense
• -k = we

That is why Turkish does not need the subject pronoun biz here. The verb already shows that the subject is we.

Because Turkish often drops subject pronouns when the verb already makes the person clear.

• ettik already tells you the subject is we

So:

• Kayıt ücreti ile geçerlilik süresini birlikte kontrol ettik.
• Biz kayıt ücreti ile geçerlilik süresini birlikte kontrol ettik.

Both are possible, but the version without biz is more natural unless you want emphasis.

No, Turkish word order is flexible, although the neutral pattern usually puts the verb at the end.

Your sentence follows a very normal order:

• object phrase
• adverb
• verb

So:

• Kayıt ücreti ile geçerlilik süresini birlikte kontrol ettik.

is natural and neutral.

You could move things around for emphasis, but the verb usually stays at the end in standard statements. For example, changing the position of birlikte may shift the emphasis slightly, but not the basic meaning.