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.
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.
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.)