İlk yardım çantasını açıp bandajı çıkardık.

Because ilk yardım is the fixed Turkish expression for first aid.

• ilk = first
• yardım = help / aid

So ilk yardım çantası means first aid kit. Even though birinci can also mean first in some contexts, birinci yardım is not the normal expression here.

This form contains several parts:

• çanta = bag
• çanta-sı = its bag / the bag of ...
• çanta-sı-nı = that noun in the accusative case

In ilk yardım çantası, Turkish uses a noun compound structure. Literally, it is something like the bag of first aid, but in natural English we say first aid kit.

Then, because this whole thing is the definite direct object of açıp / açmak, it takes the accusative ending:

• ilk yardım çantası = first aid kit
• ilk yardım çantasını = the first aid kit

The n before -ı is a buffer consonant used after the possessive ending -sı.

Because bandajı is the definite direct object of the verb çıkardık.

• bandaj = a bandage / bandage
• bandajı = the bandage

In Turkish, a specific or definite direct object usually takes the accusative ending. So:

• Bandaj çıkardık would sound more like we took out a bandage
• Bandajı çıkardık = we took out the bandage

Here, the sentence refers to a particular bandage, so bandajı is natural.

Açıp comes from açmak = to open.

The ending -ıp / -ip / -up / -üp connects verbs and often means something like:

• and
• and then
• by doing

So açıp bandajı çıkardık means:

• opening it, we took out the bandage
• more naturally: we opened it and took out the bandage

The exact vowel changes by vowel harmony. Since açmak has the vowel a, the form is açıp.

Both are possible, but açıp ... çıkardık is often more natural and compact in Turkish when:

• the same subject does both actions
• the actions happen in sequence

So:

• İlk yardım çantasını açıp bandajı çıkardık.
  = We opened the first aid kit and took out the bandage.

This sounds smooth and connected.

You could also say:

• İlk yardım çantasını açtık ve bandajı çıkardık.

That is grammatically correct too, but slightly more explicit and a bit less streamlined.

No. In this sentence, açıp does not carry the main tense or person information.

The tense and person are shown by the final finite verb:

• çıkardık = we took out

So the whole sentence is understood as past tense because of çıkardık. The earlier verb açıp is linked to it and gets its time reference from the main verb.

Çıkardık can be broken down like this:

• çıkar- = take out / remove
• -dı- = past tense
• -k = we

So çıkardık means we took out or we removed.

This is from the verb çıkarmak, not çıkmak.

This is a very common learner question.

• çıkmak = to go out / come out / exit
  This is usually intransitive, so the thing comes out by itself.
• çıkarmak = to take out / remove / make come out
  This is transitive, so someone removes something.

Examples:

• Bandaj kutudan çıktı.
  = The bandage came out of the box.
• Bandajı kutudan çıkardık.
  = We took the bandage out of the box.

In your sentence, we are doing the action to the bandage, so Turkish uses çıkarmak.

Because Turkish normally prefers Subject-Object-Verb order.

Here the structure is roughly:

• İlk yardım çantasını = the first aid kit
• açıp = opening / and opening
• bandajı = the bandage
• çıkardık = we took out

So each object appears before the verb it belongs to. This is very normal in Turkish.

English says:

• We opened the first aid kit and took out the bandage.

Turkish naturally says:

• First aid kit-the opening, bandage-the took out-we.

Of course, Turkish word order is flexible, but this version is very standard.

Yes. The subject is not stated separately because it is already built into the verb çıkardık.

The ending tells us the subject is we.

So Turkish does not need to say biz unless there is special emphasis.

• Çıkardık = we took out
• Biz çıkardık = we took out, with extra emphasis on we

That is why the sentence can simply be:

• İlk yardım çantasını açıp bandajı çıkardık.

Yes. That is possible, depending on context.

• çıkarmak can mean take out, remove, or sometimes pull out
• bandajı çıkardık could mean we took out the bandage if it was inside the kit
• in some contexts, bandajı çıkardık could also mean we removed the bandage from somewhere

But with ilk yardım çantasını açıp, the most natural interpretation is:

• We opened the first aid kit and took out the bandage

because the bandage is presumably inside the kit.

Yes, if the context already makes it clear what was opened.

For example:

• Açıp bandajı çıkardık.
  = We opened it and took out the bandage.

This works if listeners already know what it is.

But in a standalone sentence, ilk yardım çantasını is helpful because it makes the meaning complete and clear.