Which one is grammatically correct?

I’m not entirely sure which one is grammatically correct. However, in each example, it seems to me that the word “being” is entirely unnecessary and strange. Take out that word and they seem fine to me. (“The error message shown on the Home page is wrong.”)

I am an idiot, so I expect to be corrected here.

First example: Either use is technically correct, but both are somewhat wordy and awkward. I’d suggest:
“The error message currently shown on the home page is wrong.” (I’d say what is wrong: “shows incorrect results / reasons” or “is out of date” or whatever the specific problem is.)

None of the other examples is correct.

Suggestion for the second example:
Old Default Display pictures are being used on the site. (Even this is awkward.) Better would be: The site has old Default Display photos.

I can’t even understand what the last example is attempting to say. Is it “We have not set a mandatory rating for books listed under ‘My Books’ category.”? (I still can’t make much sense of it.)

@tom_g : I am just referring this one.

The error message shown on the home page is wrong (actually I would use incorrect instead of wrong most likely)

Old Default Display pictures are used on the site.

We have not yet set the ratings mandatory for books listed under the “My books” category.

@JLeslie : The sentence you have described here indicate simple present. Comparing them with the above sentences the difference is those are Present continuous.

Also, make the first sentence like,
The error message displayed on the home page is wrong.

@WasCy : Yes. The third sentence means “We have not set a mandatory rating for books listed under ‘My Books’ category.

I think, the only thing is the tense used here. Correct me if I am wrong

Some other examples are:
Any chance of being selected?
I hate being told the same thing twice.

Then just use being, don’t add in the that. The error message being shown….

The “that” is dead wood in the sentence in my opinion.

With the first and third examples, both versions are correct. The difference is that the first version uses a reduced subordinate clause and the second version uses a full subordinate clause. The meaning is the same for both versions.

You can reduce a subordinate clause, but you can’t reduce a main clause. This is why, with the second example, the first version is wrong. The verb for the main clause (“is”) is missing.

The error message being shown on the Home page is wrong.
Old Default Display pictures is being used on site.