kyou no sindan de ha, ookina mondai ha nai rasii.

Here, 今日 is functioning like a noun, and の links it to another noun, 診断.

So:

• 今日 = today
• 診断 = diagnosis, medical assessment, examination result
• 今日の診断 = today's diagnosis / the diagnosis from today

This is the same pattern as:

• 昨日のニュース = yesterday's news
• 日本の車 = Japanese cars / cars from Japan

では is で + は.

• で can mark the setting, basis, or source
• は adds a topic or contrast nuance

So 今日の診断では means something like:

• in today's diagnosis
• according to today's diagnosis
• based on today's examination

The は makes it sound a bit like setting the frame:

• As for today's diagnosis...

It can also imply a slight contrast, such as at least from today's diagnosis.

Japanese can have more than one は in a sentence, because they can mark different layers of topic or contrast.

In this sentence:

• 今日の診断では → sets the overall context: as for today's diagnosis
• 大きな問題は → highlights major problems as the thing being discussed

So the structure is roughly:

• As for today's diagnosis, as for any major problems, there apparently aren't any.

This sounds natural in Japanese, even though it feels unusual from an English point of view.

Both 大きな問題 and 大きい問題 are possible.

In this sentence, 大きな問題 sounds very natural and common. It often has a slightly more set, qualitative, or subjective feel than 大きい問題.

A rough way to think about it:

• 大きい問題 = a big problem, more directly descriptive
• 大きな問題 = a major/significant problem, often a bit more natural in set phrases

So 大きな問題はない is a very common expression for there is no serious problem or there are no major problems.

In this sentence, らしい means apparently, it seems, or I hear that.

So ないらしい means:

• apparently there isn't
• it seems there isn't

The important nuance is that the speaker is not presenting it as a direct, absolute statement of fact. They are basing it on:

• what someone said
• what they heard
• what the results seem to indicate

So compared with:

• 大きな問題はない = there is no major problem
• 大きな問題はないらしい = apparently there is no major problem

らしい normally attaches to the plain form.

So:

• plain negative: ない
• • らしい
• = ないらしい

If you want to make the whole sentence polite, you usually add politeness at the end:

• 大きな問題はないらしいです。

That is much more natural than trying to attach らしい to a polite form like ありません.

Japanese nouns usually do not show singular vs. plural clearly unless the context makes it necessary.

So 問題 can mean:

• a problem
• problems

In this sentence, English will often translate it as plural:

• There apparently aren't any major problems.

But singular is also possible depending on context:

• There apparently isn't any major problem.

Japanese leaves that open.

Yes. Japanese often leaves out subjects and other information when they are understood from context.

This sentence does not explicitly say:

• whose diagnosis it is
• who was examined
• what exactly is being discussed

Depending on context, it could mean:

• In today's diagnosis, I apparently don't have any major problems.
• According to today's exam results, there apparently aren't any major issues.
• From today's medical assessment, it seems there is no major problem.

This kind of omission is very normal in Japanese.

A useful breakdown is:

• 今日の = today's
• 診断では = in/according to the diagnosis
• 大きな問題は = as for major problems
• ない = there are not / does not exist
• らしい = apparently / it seems

A very literal English-style parsing would be:

• As for today's diagnosis, as for major problems, apparently there are none.

That is not natural English, but it helps show how the Japanese sentence is built.

Yes, it sounds natural.

It has a calm, slightly indirect tone because of らしい, which is very common when the speaker is reporting or cautiously interpreting medical information.

Without らしい, it would sound more definite:

• 今日の診断では、大きな問題はない。

With らしい, it becomes softer and less absolute:

• 今日の診断では、大きな問題はないらしい。

That makes good sense in a medical context, especially when someone is passing on results rather than making a firm expert statement themselves.