Ben anahtarımı bulamıyorum, bu yüzden kapıyı açamıyorum.

Yes. In Turkish, the ending -um in bulamıyorum and açamıyorum already tells you the subject is I.

So Ben is optional here. It may be used:

• for emphasis
• for clarity
• for contrast, like I can’t find my key

A very natural version is also:

Anahtarımı bulamıyorum, bu yüzden kapıyı açamıyorum.

Anahtarımı breaks down like this:

• anahtar = key
• -ım = my
• -ı = accusative marker, showing it is the specific direct object

So:

• anahtarım = my key
• anahtarımı = my key, as the specific thing being affected by the verb

In this sentence, it is the thing being searched for, so it takes the object marking.

Because both are specific direct objects.

In Turkish, a direct object is often marked with the accusative suffix when it is:

• definite
• specific
• known
• already identified in context

Here:

• anahtarımı = my key, a specific key
• kapıyı = the door, a specific door

This is different from a more general or non-specific object. For example:

• kitap okuyorum = I am reading a book / books
• kitabı okuyorum = I am reading the book

Because bulamıyorum means I can’t find, not just I’m not finding.

This is an important difference:

• bulmuyorum = I am not finding / I am not looking for it in that way / I’m not finding any
• bulamıyorum = I cannot find it

In Turkish, ability is often shown with -abil- / -ebil- in the positive:

• bulabiliyorum = I can find

But in the negative, Turkish usually drops -bil- and uses:

• bulamıyorum = I can’t find

So this is a very common pattern:

• yapabiliyorum = I can do it
• yapamıyorum = I can’t do it

It is the present continuous / progressive ending.

So literally, the verbs are in a form like:

• I am not able to find
• I am not able to open

But in natural English, they are usually translated as:

• I can’t find
• I can’t open

Turkish often uses this form for a current, ongoing situation.

Bu yüzden means:

• therefore
• for this reason
• so
• that’s why

It connects the first clause to the second:

• I can’t find my key, so I can’t open the door.

A very common spoken alternative is o yüzden, which also means so / that’s why.

The y is a buffer consonant.

Here is what happens:

• kapı ends in a vowel
• the accusative ending also begins with a vowel: -ı

Turkish does not usually like two vowels meeting directly, so it inserts y:

• kapı + ı → kapıyı

This same thing happens in many Turkish words:

• arabayı
• masayı
• odayı

They mean different things:

• açamıyorum = I can’t open it
• açmıyorum = I’m not opening it / I don’t open it

So:

• açamıyorum expresses inability
• açmıyorum expresses simple negation

In this sentence, the reason is that the speaker does not have the key, so inability is the correct meaning.

Yes, Turkish word order is flexible, although the verb often stays near the end.

This sentence uses a very normal structure:

• Ben anahtarımı bulamıyorum
• bu yüzden kapıyı açamıyorum

You can also omit Ben:

• Anahtarımı bulamıyorum, bu yüzden kapıyı açamıyorum.

That is probably the most natural everyday version.

Turkish changes word order more easily than English, often to shift emphasis.

Vowel harmony helps determine which vowel appears in a suffix.

For example:

• anahtar has the last vowel a, so you get -ım and -ı

  • anahtarım
  • anahtarımı

• kapı has the last vowel ı, so the accusative becomes -ı

  • kapıyı

• açamıyorum has a in the stem, so the negative/ability-related part is -ama-, not -eme-

A simple way to think about it: Turkish suffix vowels usually adjust to match the word before them.

That is why suffixes may appear as:

• ı / i / u / ü
• a / e

depending on the vowels in the base word.