Kapının yanındaki ayakkabılar benim.

Because kapının is the genitive form, meaning of the door.

• kapı = door
• kapının = of the door / the door’s

In this sentence, Turkish is building a phrase like the one at the side of the door.
So kapının tells you that yan (side) is related to the door.

A very literal breakdown is:

• kapı = door
• kapının yanı = the side of the door
• kapının yanındaki ayakkabılar = the shoes at the side of the door

So the -ın ending on kapı is there because Turkish uses the genitive to connect door with side.

Yanındaki means something like the one(s) next to / beside / at the side of.

It comes from several parts:

• yan = side
• -ı / -i = a possessive ending here, giving yanı = its side / the side
• -nda / -inde / -ında type location idea = in/on/at
• -ki = a suffix that turns the location phrase into something like the one that is at...

So:

• kapının yanı = the side of the door
• kapının yanında = at the side of the door / next to the door
• kapının yanındaki = the one(s) that is/are next to the door

Before a noun, -ki phrases work like adjectives:

• kapının yanındaki ayakkabılar = the shoes that are next to the door

So yanındaki is doing the job that English often does with that are next to.

Because Turkish often uses a modifier before the noun instead of a separate relative clause.

English says:

• the shoes that are next to the door

Turkish packages this more tightly:

• kapının yanındaki ayakkabılar
• literally: the next-to-the-door shoes

This is very normal in Turkish. Instead of using a separate word like that plus a verb like are, Turkish often uses endings and suffixes to create a noun modifier.

So yanındaki already includes the idea of:

• that is / that are
• located next to

Yes, ayakkabılar means shoes, and in this sentence it is naturally understood as the shoes.

• ayakkabı = shoe
• ayakkabılar = shoes

Turkish does not have a separate word like the.
Whether something is understood as a shoe, the shoe, or shoes / the shoes depends on context.

Here, because the phrase is specific:

• kapının yanındaki ayakkabılar = the shoes next to the door

So even though there is no separate word for the, the phrase is definite because it points to a specific set of shoes.

In this kind of sentence, benim can mean mine or belong to me.

• ben = I
• benim = my / mine (depending on use)

In Kapının yanındaki ayakkabılar benim, the meaning is:

• The shoes next to the door are mine
• literally closer to: ... are of me / belong to me

This is a very common Turkish pattern:

• Bu kitap benim. = This book is mine.
• Bu ev bizim. = This house is ours.

So although benim often means my before a noun, it can also stand at the end of a sentence and mean mine.

In this sentence, benim is the natural choice.

• Ayakkabılar benim. = The shoes are mine.

Benimki usually means mine one / my one / the one that is mine, and it is often used when the noun is omitted and understood from context.

For example:

• Senin ayakkabıların siyah, benimkiler beyaz.
  = Your shoes are black, mine are white.

Or:

• Benimki kapının yanında.
  = Mine is next to the door.

So:

• ayakkabılar benim = correct and natural
• ayakkabılar benimki = not right here

If you wanted a form with -ki, it would usually need to match the plural idea differently, and even then benim is still the standard way in this kind of ownership sentence.

Turkish often leaves out the verb to be in the present tense.

So:

• Ayakkabılar benim. literally looks like:
• The shoes mine.

But it means:

• The shoes are mine.

This is normal Turkish grammar. In present-tense equational sentences, the copula (am / is / are) is often not written as a separate word.

Compare:

• Ben öğrenciyim. = I am a student.
• O mutlu. = He/She is happy.
• Bunlar benim. = These are mine.

So the are is understood, even though it is not a separate word.

Because in Turkish, modifiers usually come before the noun they describe, and the predicate often comes at the end.

Structure:

• Kapının yanındaki = next to the door
• ayakkabılar = shoes
• benim = mine

So the order is:

1. descriptive phrase
2. noun
3. predicate

That gives:

• Kapının yanındaki ayakkabılar benim.

English prefers:

• The shoes next to the door are mine.

Turkish prefers to place the descriptive part before ayakkabılar.

Yes. A common alternative is:

• Kapının yanındaki ayakkabılar bana ait.

This also means:

• The shoes next to the door are mine
• more literally: The shoes next to the door belong to me

Difference in feel:

• ... benim = simple, everyday, very common
• ... bana ait = also common, slightly more explicit about ownership

Both are correct.

Yes. This pattern is very productive.

Examples:

• Kapının yanındaki masa = the table next to the door
• Kapının yanındaki adam = the man next to the door
• Kapının yanındaki oda = the room next to the door

So once you learn X-in yanındaki Y, you can use it with many nouns:

• evin yanındaki park = the park next to the house
• okulun yanındaki kafe = the café next to the school

It is a very useful structure for location.

Yes. A helpful literal breakdown is:

• kapının = of the door
• yanındaki = the one(s) at its side / next to it
• ayakkabılar = shoes
• benim = mine / belong to me

So a very literal reading is:

• The shoes at the side of the door are mine.

That is not the most natural English translation, but it shows how the Turkish sentence is built.