mauseuga chaeksang araero tteoreojyeoseo orae chajasseo.

In most everyday Korean, 마우스 usually means a computer mouse. If someone means the animal, they more often say 쥐.

So in this sentence, the most natural interpretation is a computer mouse falling under the desk.

가 is the subject marker. It marks 마우스 as the thing that fell.

So:

• 마우스가 떨어졌어 = The mouse fell

If you used 마우스는, it would sound more like you are setting up the mouse as the topic, possibly with a contrastive nuance:

• 마우스는 책상 아래로 떨어져서... = As for the mouse, it fell under the desk...

In this sentence, 가 is the most neutral and natural choice.

책상 아래 means under the desk.

The particle 로 often marks direction or the destination/resulting direction of movement. So:

• 책상 아래로 떨어지다 = to fall down under the desk

It gives the sense of moving to/toward the space under the desk.

So the phrase is not just under the desk, but more like down to under the desk.

Yes, you could, and both are natural.

The difference is small:

• 책상 아래로 떨어졌어: focuses a bit more on the direction/path of the fall
• 책상 아래에 떨어졌어: focuses a bit more on the location where it ended up

In everyday conversation, both are possible here.

It comes from the verb 떨어지다, which means to fall.

Here is the breakdown:

• 떨어지다 → stem 떨어지-
• add -어서
• 떨어지어서 contracts to 떨어져서

So 떨어져서 means something like:

• because it fell
• after it fell
• having fallen

In this sentence, it naturally connects the first event to the second: the mouse fell under the desk, so the speaker spent a long time looking for it.

It can feel like both.

-아/어서 often connects two actions or situations where the first leads naturally to the second. In this sentence:

• the mouse fell under the desk
• as a result, I searched for a long time

So the most natural understanding is because it fell under the desk, I searched for a long time.

A simple and then translation is possible, but cause/result is stronger here.

Yes. Korean often omits subjects when they are understood from context.

So the sentence works like this:

• 마우스가 책상 아래로 떨어져서 = the mouse fell under the desk
• (내가) 오래 찾았어 = (I) looked for it for a long time

Even though the first clause has the mouse and the second clause has an implied I, this is completely natural in Korean when the meaning is clear.

Because Korean very often drops obvious objects.

The thing being searched for is clearly the mouse, so it does not need to be repeated.

A fuller version would be:

• 마우스가 책상 아래로 떨어져서 마우스를 오래 찾았어.

But that sounds more repetitive. The original sentence is more natural.

This is a great question, because 찾다 can mean either:

• to look for / search for
• to find

In this sentence, because of 오래 (for a long time), the natural reading is looked for or searched for.

So:

• 오래 찾았어 = I looked for it for a long time

It may imply that the speaker eventually found it, but that is not stated clearly.

If you wanted to make the successful finding explicit, you could say something like:

• 오래 찾다가 결국 찾았어. = I looked for it for a long time and finally found it.

오래 means for a long time.

It modifies 찾았어, so:

• 오래 찾았어 = searched for a long time

Yes, 오랫동안 can also be used:

• 오랫동안 찾았어

Both are natural. 오랫동안 sounds a little more explicit or slightly more formal, while 오래 is very common and conversational.

찾았어 is the casual/informal speech style.

So this sentence sounds like something you would say to:

• a friend
• someone younger
• someone you are close to

If you want to make it polite, you would say:

• 마우스가 책상 아래로 떨어져서 오래 찾았어요.

The meaning is the same; only the speech level changes.