uisaga sajineul boyeo jumyeonseo chungchiga du gae issdago haesseoyo.

Both are possible, but they feel a little different.

• 의사가 presents the doctor as the subject of this specific action.
• 의사는 would give more of a topic feeling, like as for the doctor...

In this sentence, 의사가 is very natural because the speaker is simply describing what happened:

• 의사가 사진을 보여 주면서... 했어요
  = The doctor, while showing the picture, said...

So -가 here is just marking the subject in a straightforward way.

Because 사진 is the direct object of 보여 주다.

• 사진 = photo / image / X-ray / picture
• 보여 주다 = to show someone something

So 사진을 보여 주면서 means:

• while showing the picture

The object particle -을/를 marks what is being shown.

보여 주면서 comes from 보여 주다, which means to show.

It is made of:

• 보이다 / 보여 주다 = to show
• -면서 = while doing

So:

• 사진을 보여 주면서 = while showing the picture

The nuance is that the doctor was showing the picture and speaking at the same time.

Because 보다 means to see / to look at, while 보여 주다 means to show.

Compare:

• 사진을 보다 = to look at a picture
• 사진을 보여 주다 = to show a picture

In this sentence, the doctor is the one showing the picture to the patient, so 보여 주다 is the correct verb.

-면서 means while doing, connecting two actions that happen at the same time.

Here:

• 사진을 보여 주면서
• 충치가 두 개 있다고 했어요

So the idea is:

• The doctor said there were two cavities while showing the picture.

It often gives a sense of two simultaneous actions by the same subject.

Korean usually counts nouns with a counter.

Here:

• 충치 = cavity
• 두 = two
• 개 = general counter for things

So:

• 충치가 두 개 있다 = there are two cavities

You normally need the counter:

• 한 개, 두 개, 세 개...

Without the counter, it would sound incomplete or unnatural in standard Korean.

Because Korean uses special shortened forms of native Korean numbers before counters.

So:

• 하나 → 한
• 둘 → 두
• 셋 → 세
• 넷 → 네

That is why you say:

• 두 개 not
• 둘 개

This is a very common pattern in Korean.

Here 충치가 두 개 있다 is the content of what the doctor said.

Inside that clause:

• 충치가 = cavities are / as for cavities
• 두 개 있다 = there are two

So -가 marks 충치 as the subject of the existence verb 있다.

This is very common with 있다:

• 문제가 있어요 = there is a problem
• 사람이 많아요 = there are many people
• 충치가 두 개 있어요 = there are two cavities

This is reported speech, often called indirect quotation.

It breaks down like this:

• 있다 = to exist / to have
• -다고 = quotation form used after descriptive verbs and 있다/없다
• 했어요 = said

So:

• 충치가 두 개 있다고 했어요 = (He/She) said that there were two cavities

This is one of the most common ways to report what someone said in Korean.

Because -다고 하다 is used after:

• descriptive verbs
• 있다 / 없다
• nouns with 이다 in some patterns

Since 있다 is the verb here, -다고 하다 is correct:

• 있다 → 있다고 했어요

By contrast, -라고 하다 is often used after nouns for quoted statements like naming or calling something something.

So in this sentence, 있다고 했어요 is the normal form.

Because the act of saying happened in the past.

The sentence is not directly stating there are two cavities as the speaker’s own claim. It is saying:

• The doctor said there were two cavities.

So the final verb reflects the completed speech event:

• 했어요 = said

Even if the cavities may still exist now, the doctor’s statement happened in the past, so 했어요 is natural.

Yes, in everyday writing you may also see 보여주다 and 보여주면서.

So these are both commonly seen:

• 보여 주다
• 보여주다

And:

• 보여 주면서
• 보여주면서

Spacing with auxiliary-like verbs in Korean can vary in real usage. Learning materials often separate them to make the structure easier to see, but both spellings appear.

A helpful way to see it is:

• 의사가 = the doctor
• 사진을 보여 주면서 = while showing the picture
• 충치가 두 개 있다고 했어요 = said that there were two cavities

So the overall structure is:

• [subject] + [while doing X] + [said that Y]

That makes the sentence easier to unpack:

1. The doctor was showing a picture.
2. At the same time, the doctor said something.
3. What the doctor said was: there are two cavities.

No. 두 개 is natural and understandable, but in real Korean people may also say 두 군데 depending on the nuance.

Compare:

• 충치가 두 개 있어요 = there are two cavities
• 충치가 두 군데 있어요 = there are cavities in two places / two spots

The sentence you were given treats cavities as countable items, so 두 개 is completely fine.

Yes, very often Korean drops subjects when they are clear from context.

So you might hear:

• 사진을 보여 주면서 충치가 두 개 있다고 했어요.

This would still be understood as something like:

• (The doctor) said there were two cavities while showing the picture.

Korean often omits subjects if the listener can figure them out.

Yes. Korean usually puts background information before the main verb.

So:

• 사진을 보여 주면서 comes first as the situation/background
• ...했다고 했어요 comes at the end as the main action

This is very typical Korean word order. English often says the main action earlier, but Korean builds up to it, with the final verb carrying the core statement.