1. How long does a black hole exist? Black holes grow as the acquire mass from infalling material and shrink slowly through Hawking radiation. A BH will 'evaporate' when it can no longer gain mass. How long that takes will of course depend on the mass of the BH and the rate of evaporation. 2. How long did the singularity exist before the big bang? Time itself did not exist before the big bang. You can say the singularity *always* existed, and that it existed for no time at all, and be correct both ways.
Similarly, one might say either that the north pole exists everywhere greater than or equal to 90 degrees north latitude or that the north pole exists only at 90 degrees north latitude. Since there's nothing north of the north pole, both are equivalent.
Unknown. There may not have even been a singularity, and I know one of our Cosmology experts claims that practically no professional Cosmologist actually believes that the singularity predicted in the theory is real. Instead it is believed to be the result of our incomplete knowledge of how physics works at that scale.
There WAS time before the big bang, but it is extremely insignificant. Read on A Brief History of Time by Stephen Hawking which delves into this very concept.
when a singularity can exist at the centre of a black hole ( this is what rigorous mathematics proves), what can't it exist before the big bang?
Those rigorous mathematics are most likely wrong. It is probable that we don't know how physics works at the scale of black holes, causing our math to cease to accurately represent reality. Remember, there are no "proofs" when it comes to describing the real world. Only pure mathematics has proofs.
Nonsense. "Rigorous mathematics" cannot prove anything about physics. And I doubt that any intelligent mathematician would want to express an opinion as to what a mathematical "singularity" would mean physically.
No one should believe the result of some Maths without testing its validity. Even the simplest SUVAT calculations can give you an impossible answer if you don't take the right sign for the square root.