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What would you say when unexpectedly finding yourself in big trouble while everything itself is not going very well and an irrevocable, unpleasant happening takes place suddenly in the manner that you immediately notice that a big problem / a series of problems are about to begin. Please consider my self-made scenario below:

• 
  Son) There is a long distance between my apartment and the university where the conference would be held!
  
  I must leave at 2 to make a good time to the conference; it's 1 o'clock and I'm running out of time!
  Mother) So, whats wrong? Come on son! Move right now!
  Son) Oh, ..........................
  Mother) Why?! What's the matter again?
  Son) I just noticed that my car is broken!

In our language, there is a proverb to be used to fill in the blank which says:

• My cow gave birth.

Even sometimes, in order to exagerate the situation we say:

• "My cow has given birth to twin calves"!

Here is the outcome of my researches in this regard, which I have no any ideas whether using each one sounds natural and idiomatic in this sense! Meanwhile, I don't know if they are considered as old-fashioned expressions these days or somehow, some translated versions of a similar foreign saying:

• That's the pretty kettle of fish!
  


• That fat in the fire!


• Here comes the trouble!


• Our goose is cooked!


• That crowns it all.


• That's the end.

I would be appreciative if you could help me to find the closest idiom/expression/proverb for this case.

Of the suggested expressions "That crowns it" is an ironic way of saying that something is the thing that finishes off the problem. That seems the most appropriate expression in this situation.

A similar, but slightly different meaning is "That's the straw that broke the donkey's back", meaning the small thing that wouldn't be a big problem, except it came after lots of other problems.

But using these proverbs is not very natural. When native speakers use them it is usually referring to the proverb, not just using it.

They talk about "the straw that broke the donkey's back". Well ...

In this situation, the language that people use would probably be a lot coarser.

Mom: Come on son, move right now!

Son: Oh Bugger!

Mom: What's the matter?

There are lots of other coarse words, some of which you might not want to say in front of you mother, depending on how sensitive she is. (Bugger is a British only expression)

Or you could say something like "That's all I need!"

To describe the situation you might say "I'm up the creek without a paddle" But this is usually "I'm up shit creek (without a paddle)". You might also say that the person is "screwed". So if we put it together:

Bugger. Well I'm really up shit creek now. The conference is in two hours, there's an hour's drive to the university and to crown it all my car won't start. I'm basically screwed.

There are some minor issues with the examples you suggested. Here are revised versions, with comments:

That's a pretty kettle of fish!

For no clear reason, this is always "a", never "the" It is also a bit old-fashioned. I don't recall the last time I herd this outside of a novel set quite a while ago, say the early 1900s or before.

The fat is in the fire!

Always said in these words, never 'that fat" and never without the "is" or a contracted "is" in the form of "'s". Also a bit old-fashioned, not really current usage.

Here comes trouble!

Never with "the". Usually indicates a specific problem beleived to be imminent, or the approach of a person who is likely to cause a problem. Not used for the accumulation of several problems as described in the question.

Our goose is cooked!

Can also be "my goose" or "your goose". A bit old-fashioned, but still in current use.

That crowns it all.

Not in current use, and not to my knowledge ever in really wide use. I would avoid this.

That's the end.

Perfectly acceptable, but not really a proverb or traditional saying.

At least in US English, the use of proverbs or traditional sayings as idioms is much less common than it was, say, 100 years ago.

People now might say:

Oh, What a mess!

We've got serious problems here!

(or "a serious problem here")

Houston, we've got a problem

Refers to the typically understated phrasing used by astronauts to report a very serious possibly life-threatening, issue. Mission control for all US manned space travail was in Houston. This phrase became iconic.

The world just landed on me.

That's the last straw.

Implies an accumulation of problems, and the latest one is just too much. alludes to 'The straw that broke the camel's back" the final tiny increase of load which causes catastrophe.