It's technically possible to get every letter on the first try.
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That depends on how the random number generator works. Generally there's a relatively small number of seeds you can start with, and if there's nothing the game uses rng for besides the numbers the monkeys pick it will always pick the same series of letters from the same seed, so regardless of what seed it picks it won't be able to get all the letters on the first try.
theoretically, yes, assuming the character order isn't predetermined. if it is not predetermined, each character has a 1/36 (2.778%) chance of being the correct one. then the next character has a 1/36 chance. and so on. so the odds of getting every single letter on the first try would be 1 in some trillions or something. i don't feel like counting how many letters spaces and symbols are in act 1 scene 1 of king lear. if it is predetermined, i have no freaking clue what the odds would be.
guh i came back to this game over 2 years later and realized by reading the text below the game that it's really 1/34 and it's 14326 characters long. using a bit of maaaath that's... 1 in a really big number that rounds to a 9 with 21939 zeroes. i wanted to put the whole number but it exceeds the character limit of 20,480 lol