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My main question is: Can the singular forms and plural forms be used interchangeably after the word "no"?
For example, are these sentences interchangeable?
1) "I understood no words he said."
2) "I understood no word he said."
Another example: Are these sentences interchangeable?
3) "There are no black students in his school."
4) "There is no black student in his school."
No, they are not interchangeable.
Both the sentences:
*There is no snake in Ireland,
*There were no moons in the sky,
are completely unnatural. You need to say:
There are no snakes in Ireland,
There was no moon in the sky.
Why? You use the singular when you expect there to be only one of something, and you use the plural when you expect there to be more than one.
Of course, often the number you expect is less clear than in these two cases, and you can use either singular or plural. For your two sentences, I would use
I did not understand any words he said, or
I did not understand a single word he said,
There are no black students at his school.
Why not use understood no word or words in the first example? I'm not sure; maybe just because that isn't the way we usually phrase it. Googling, understood no word and understood no words are both used.
"understood no word", there's still a few pages more than for the plural alternative (see Google Books Ngram Viewer). However, Practical English Usage (M. Swan) puts it this way: "After no, countable nouns are usually plural unless the sense makes a singular noun necessary."