Kat mülkiyeti arsa payını içeriyor.

Yes. In kat mülkiyeti, the -i is the third-person possessive ending, but here it is being used as part of a very common Turkish noun compound pattern.

Breakdown:

• kat = floor / storey / apartment unit
• mülkiyet = ownership
• kat mülkiyeti = literally something like floor/apartment ownership

In Turkish, compounds of the type X ownership, X door, X price, etc. are often formed like this:

• first noun: bare
• second noun: takes a possessive-type ending

So kat mülkiyeti does not necessarily mean his/her ownership of the floor. It is just the normal compound form.

The base form is arsa payı.

Breakdown:

• arsa = land / plot
• payı = share/portion (in compound form)

So:

• arsa payı = land share / share of the land

Then the sentence uses arsa payını, which is:

• arsa payı
  • -(y/n)ı accusative

More exactly here, it becomes payı + nı, giving arsa payını.

That n is a buffer consonant.

The noun payı already ends with a possessive ending (-ı as part of the compound). When Turkish adds another ending after that, it often inserts n to make pronunciation and structure smoother.

So the form is:

• arsa payı = land share
• arsa payı-n-ı = the land share (as a definite direct object)

This buffer n is very common after possessive forms.

Because it is the definite direct object of the verb içeriyor.

The verb is:

• içermek = to include / contain

What does the ownership include?

• arsa payını = the land share

In Turkish, a specific/definite direct object usually takes the accusative. That is why you get -ı / -i / -u / -ü here.

So:

• arsa payı = a land share / land share in a general sense
• arsa payını = the land share / its land share, a specific one

Because Turkish normally uses Subject–Object–Verb word order.

In this sentence:

• Kat mülkiyeti = subject
• arsa payını = object
• içeriyor = verb

So the order is completely normal for Turkish.

English usually prefers:

• Subject + Verb + Object
• Condominium ownership includes the land share

Turkish usually prefers:

• Subject + Object + Verb
• Kat mülkiyeti arsa payını içeriyor.

Both can be possible, but they feel a little different.

• içeriyor = literally is including / includes
• içerir = includes in a more general, formal, or timeless sense

In modern Turkish, -iyor is often used even for general facts, especially in everyday language. So içeriyor does not have to sound as temporary as English is including.

Very roughly:

• Kat mülkiyeti arsa payını içeriyor. = natural, everyday
• Kat mülkiyeti arsa payını içerir. = also correct, a bit more formal or definition-like

Because Turkish has no articles like English a/an/the.

Whether something is understood as a, the, or just a general noun depends on context and grammar.

For example:

• kat mülkiyeti can mean condominium ownership / the condominium ownership, depending on context
• arsa payını is marked with the accusative, so it usually refers to a specific land share

So Turkish does not need separate words for the or a in the way English does.

Because both are part of Turkish noun compounds.

This is very common. Compare:

• ev kapısı = house door
• okul bahçesi = school garden
• arsa payı = land share
• kat mülkiyeti = condominium/apartment ownership

In these structures, the second noun often takes a possessive-looking ending, but the whole phrase is not always truly possessive in the English sense. It is simply how Turkish builds many compound nouns.

So in this sentence:

• kat mülkiyet-i
• arsa pay-ı

both follow that pattern.

Yes, but the neutral order is the one you were given.

Normal, neutral:

• Kat mülkiyeti arsa payını içeriyor.

You could also change the order for emphasis, for example:

• Arsa payını kat mülkiyeti içeriyor.

That would sound more marked, as if emphasizing it is the condominium ownership that includes the land share.

So for most learning purposes, the original sentence is the safest and most natural pattern to remember.