Какая находка!

Because находка is a feminine singular noun, and какая has to agree with it.

In Russian, words like what a ...! use a form that matches the noun's:

• gender
• number
• case

Since находка is feminine singular, you get:

• какой
  • masculine noun
• какая
  • feminine noun
• какое
  • neuter noun
• какие
  • plural noun

So:

• Какой подарок! = What a gift!
• Какая находка! = What a find!
• Какое место! = What a place!
• Какие люди! = Well, look who's here! / What people!

Here какая is an exclamatory word, not really a normal question word.

In a question, какая? means which? / what kind of?

For example:

• Какая книга тебе нравится? = Which book do you like? / What kind of book do you like?

But in Какая находка!, it means something more like:

• What a find!
• What an amazing find!

So this is an emotional exclamation, not a request for information.

It is in the nominative case.

Even though English says What a find!, Russian uses the noun in nominative in this kind of exclamation:

• Какая идея!
• Какой день!
• Какое счастье!

So находка stays in its dictionary form: находка.

It can, but it would sound a bit different.

Both patterns are possible:

• Какая находка!
• Что за находка!

The first one is a very standard exclamatory pattern with какой / какая / какое / какие. The second one is also common and often feels a bit more expressive or conversational.

Very roughly:

• Какая находка! = What a find!
• Что за находка! = What a find! / What an incredible find!

So Какая находка! is completely natural and normal.

No. Находка can be both literal and figurative.

It can mean:

1. a thing someone has found
2. a lucky discovery
3. something or someone that turns out to be extremely useful or valuable

So Какая находка! could be said:

• when you find an old coin in the attic
• when you discover a great little café
• when someone recommends a perfect tool or app
• even about a person, in the sense of What a gem! or What a find!

The stress is:

• кака́я находка

More specifically:

• кака́я = stress on the second а
• нахо́дка = stress on хо

A rough pronunciation guide:

• ka-KA-ya na-KHOD-ka

The х is like the ch in Scottish loch or German Bach, not like English h.

Yes, but Какая находка! is the most neutral and natural order.

You may sometimes hear different orders for emphasis, but they are more marked or stylistic. For a learner, the standard pattern is best:

• Какая находка!

This same structure is very common in Russian:

• Какая красота! = What beauty!
• Какой ужас! = What a horror!
• Какое совпадение! = What a coincidence!

It is basically neutral and works well in everyday speech.

It sounds natural in conversation, writing, and narration. The emotional force depends mostly on intonation:

• said casually: Oh, nice find
• said excitedly: What a find!

So it is not slang, not especially formal, and not old-fashioned.

Yes, depending on context.

Although находка often translates as find, in some situations English may prefer:

• What a discovery!
• What a great find!
• What a gem!
• This is such a find!

So the exact English wording depends on what was found and how enthusiastic the speaker is.

A very useful pattern is:

Какой / Какая / Какое / Какие + noun!

It means What a ...! or How ...! in an exclamatory sense.

Examples:

• Какой человек! = What a person!
• Какая погода! = What weather!
• Какое утро! = What a morning!
• Какие новости! = What news!

So Какая находка! is a great example of a very common Russian exclamation pattern.