If we know the GCD and LCM of two integers, can we determine the possible values of these two integers? I know that $\gcd(a,b) \cdot \operatorname{lcm}(a,b) = ab$, so if we know $\gcd(a,b)$ and $\operatorname{lcm}(a,b)$ and we want to find out $a$ and $b$, besides factoring $ab$ and find possible values, can we find these two integers faster?

If you scale the problem by dividing through by $\rm\:gcd(a,b)\:$ then you are asking how to determine coprime factors of a product. This is equivalent to factoring integers.
Your original question, in the special case $\rm\:gcd(a,b) = lcm(a,b),\:$ is much easier:
Hint $\:$ In any domain $\rm\:gcd(a,b)\ |\ a,b\ |\ lcm(a,b)\ $ so $\rm\:lcm(a,b)\ |\ gcd(a,b)\ \Rightarrow $ all four are associate. Hence all four are equal if they are positive integers. If they are polynomials $\ne 0$ over a field then  they are equal up to nonzero constant multiples, i.e. up to unit factors.
