Does the chance of catching a Pokémon change with each throw?

Based on the quote you provided, where it says:

If Final Capture Rate is less than this random number, the Pokemon breaks out and possibly flees.

This indicates that this random number must be generated for each throw.

Many times now, I have attempted to capture Pokemon with Pokeballs, getting Great or Excellent throws, then failing, but then throwing a Pokeball with no accuracy bonus, and catching the Pokemon. If the random number did not change with each throw, these cases would be impossible.

The Random Number is generated once for each throw, the chance of catching a Pokémon does not change with each throw, except with things that explicitly do so like the Razz Berry.