In Pollard p-1 how is the bound B chosen? Does Pollard's p-1 method always produce an answer (given sufficient time and assuming input is composite)? If yes, what is the point of having the bound $B$, why not just keep increasing $a$? If no, how is $B$ reasonably guessed? Is there a limit to how large $B$ should be? For example if the input is prime then checking up $B=\sqrt{\text{input}}$ would prove there's no factors, is that correct?

The p-1-method works , if there is a prime factor $\ p\ $ of $\ N\ $ , such that $\ p-1\ $ splits into prime factors smaller than the chosen bound $\ B\ $. Whether a given bound $\ B\ $ will work, cannot be predicted.

Of course, increasing $\ B\ $ would eventually find a non-trivial factor, but at some point, this method would be not more efficient (perhaps even less efficient) then trial division.

If we are lucky, we can find large factors that could not be found with trial division in reasonable time.

The p-1-method rarely works , if there is no prime factor with , lets say , $\ 25\ $ digits or less. In this case, the better ECM (elliptic curve method) is usually used.