If you have ten thousand monkeys typing on keyboards, they'll eventually type the complete works of Shakespeare. But it will most likely happen after the universe is dead. (So they say, I'd argue it just won't happen, not in the next hundred generations.)

From a practical point of view, it's not possible to prove that one can generate all possible addresses from one seed. You can't know, because to find out, you'd have to try them all.

Just like other pseudo random number generators that start from a certain state, you can't possibly tell what was, from what's next (or the other way around) without knowing the state. Those things can have very long periods, like the Mersenne Twister.

The reason we picked such large numbers for usage in bitcoin (2^160, 2^256) is so that the odds of any two different processes coming up with the exact same private key or address is negligible or practically non-existent.

Yes, there is a chance. There's also a chance you'll win the mega millions powerball lottery or get struck by lightning several times in a row. Not just one in a million, or one in a billion. It's one in a quadrillion (or other large number, googol, age of the sun, that type of thing.)