Otdfbt

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)". It means you are in a difficult situation and it will be difficult to get out. You might also say that the person is "screwed", which just means "something bad has happened to me", but is coarse. 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 believed 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, in any and every context. I think it may once have been a Briticism, but I am not sure of that.

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 travel 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.

The phrase

I'm up Shit Creek

is na shortened version of

I am up Shit Creek without a paddle.

This is a metaphorical way of saying "I am in a very unpleasant situation, with no good way to get out of it." "Shit Creek" should be capitalized, as if it were the name of an actual place, in my view. Some consider this expression crude or impolite. It is also a bit of cliche.

"I'm screwed."

This is perfectly correct, and in wide current usage. However, many will regard this as crude, and you may wish to avoid it. "screwed" in this sense means "have had somethign bad happen to me", or "I have a prtoblem that cannot be solved." It does not directly refer to sexual activity, but that is the image being metaphorically invoked.

I'm in a fix.

means simply "I have a problem." or "I'm in a bad situation." It does not particularly imply multiple problems coming together, but can be used in such a case.

Murphy's law strikes again.

"Murphy's law" is the supposed observation, dating from the World-war II era that:

If anything can go wrong, it will.

This is often used to suggest bad luck, or that bad luck is more common than good luck.

(Oddly the original meaning was somewhat different. As I understand it the original context was an airplane part that could fit the plane in one of two mirror image positions, but would only work in one of them. The original meaning could have been rewritten as 'If something can be done incorrectly, it will be, so make sure that it can't" and the solution was to cut a notch so that the part would only fit in the correct way. All the trapazoidal cable connections and simialr devides descend from this idea.)

a big problem / a series of problems are about to begin

In regards to this part of your question, another common saying that I haven't seen mentioned is :

The Devil works in threes.

or (removing the religious aspect)

Trouble comes in threes.

There's a stack question about the origins of the "Trouble comes in threes" phrase here: death comes in threes origin. The origins of "The Devil works in threes" comes from the Biblical story of Jesus' 40 days in the wilderness where Satan made three attempts to tempt Jesus.

Typically, as others have pointed out, the blank in your statement would be filled with a curse e.g. "F***!" or blasphemy e.g. "goddammit" (or God damn it). Sometimes you might combine both options into one massive phrase.

As a bonus, there is even a phrase for thinking you have solved a problem only to end up with a much bigger problem, which is:

Out of the frying pan and into the fire.

The first expression that came to my mind was:

I'm in a bind.

This means that you're in a difficult situation that will be difficult to resolve successfully.

If you are in a situation where you have already experienced several problems, and the latest problem you encountered guarantees that you will fail, you could refer to it as:

That's the last nail in the coffin.

This is a similar expression to "That's the last straw" and indicates that the latest revelation has sealed an undesirable outcome.

Seems like it's not easy to come up with a similar idiom in English! The popular idioms I can think of tend to focus on one of two attributes of the trouble:

1. It's a small addition to your existing troubles, and it's finally pushed you personally over the edge, even though the trouble itself may not have been much.

That's the straw that broke the camel's back.

However, it sounds like you're looking for a phrase that connotes a bigger trouble — not just a "straw" but a whole bundle at once.

It never rains but it pours.

This implies that it's already been "raining" troubles on you... but here comes the metaphorical downpour. Still, this trouble didn't come out of nowhere; it's just the latest in a recent string of difficulties.

2. Trouble was pessimistically expected, and it has finally arrived.

It's always something.

Generic idioms for big-trouble-having-struck include:

That's torn it!

(Glossed as "Brit. slang: an unexpected event or circumstance has upset one's plans"; which fits your use case perfectly, but notice that this is a bit of an obscure one. The "it" in this sentence doesn't refer to any particular thing, as far as I know. It's just, whatever "it" is, it's torn now!)

The shit has hit the fan.

(Obscene/vulgar.)