It's old. Like, really old. So old that it's impossible to tell where it originated.
The book The Lore and Language of Schoolchildren by Iona and Peter Opie, published in 1959, catalogues many different schoolyard poems from throughout the first half of the twentieth century, including the following version of the one you're interested in:
Ladles and jellyspoons
I stand upon this speech to make a platform
the train I arrived in has not yet come
so I took a bus and walked
I come before you to stand behind you
and tell you something I know nothing about!
One fine day in the middle of the night
two dead men got up to a fight
back to back they faced each other
drew their swords and shot each other
a paralysed donkey passing by
kicked a blind man in the eye
knocked him through a nine inch wall
into a dry ditch and drowned them all.
I haven't actually read the Opies' book myself, but according to this page:
Opie noted that this had been collected in 12 different schools around the UK, but that it had also been collected, with almost no variation, fifty years before. It was probably older than that, too.
This rhyme most likely evolved from other nonsense rhymes based on similar themes: opposites put together, blind people watching, deaf people hearing, and so on. A manuscript in the Bodleian library dating back to 1480 contains the following lines:
I saw three headless [men] playing at a ball,
A handless man served them all.
While three mouthless men laughed,
Three legless [men] from them ran.
I've also found several sources pointing to a 1305 manuscript The Land of Cockaigne for an even older version of this kind of nonsense poem, but I haven't managed to track down a copy of the original manuscript to check the details. The primary source here appears to be The Mummers' Play by R. J. E. Tiddy, published in 1923, the relevant citation being on p. 116.
With such a long history, it seems the best conclusion we can draw is that
the idea of this poem has been around so long that it's pretty much a folk tradition. It's been handed down from generation to generation, probably mostly orally, and has of course changed and mutated quite a lot over the last seven hundred years. But trying to pin down an 'original author', for
any particular version of the poem, is a fruitless venture in this centuries-long game of Chinese Whispers. Even today, there are many slightly different versions of the poem, and no single 'canonical' wording - just look at the number of variations listed
here and
here, for example.
As for your bonus question about the genre of this poem, I would say it's a classical
nonsense rhyme. It should be noted, however, that there are two distinct kinds of nonsense poem: ones like this where the words make sense individually but become nonsensical when put together in the right way, and ones like Lewis Carroll's
Jabberwocky which are full of nonsense
words.
Some would say that only the latter counts as a true nonsense rhyme:
Although the Two Dead Boys poem (“One Fine Day in the Middle of the Night”) is often referred to as a nonsense rhyme, the description is not strictly accurate. It is clearly understandable in any of its many forms and versions and the impossibilities in the story are no more than sensible words and phrases that have been transposed. An example of a true nonsense rhyme can be seen for instance in the first four lines of Lewis Carroll’s “Jabberwocky” from Through the Looking Glass [...] One can get a feeling for the severe, gathering darkness of the poem from Carroll’s introductory lines but, until Humpty Dumpty explains it in its entirety, the poem, and particularly these first four lines, makes no sense at all.
But in everyday parlance, "nonsense verse" can certainly be used to cover both types of poem.
The former type specifically have also been called ballads of impossibilities:
The folklorist and writer Ed Cray, writing to others on an Internet ballad chat line, noted that the rhyme [the same one you're asking about] was a “Ballad of Impossibilities” and that, “A number of these songs/ballads of impossibilities were printed as broadsides in the 18th and 19th centuries.[”]