Under what condition(s) we can separate group elements from a system of relations?
For example, if we have the relations $g_{1}^{3}g_{2}^{2}g_{1}g_{2} = 1$, and $g_{2}g_{1}g_{2}g_{1} = 1$ then it is certain that we can separate $g_{1}$ and $g_{2}$ from the given two relations, and after separating them, we will have $g_{1}^{2} = 1$ and $g_{2} = 1$.
So for some relations $f_{1}, f_{2}, \ldots, f_{n}$ and for some group elements $g_{1}, g_{2}, \ldots, g_{n}$ if we have $f_{1}(g_{1}, g_{2}, \ldots, g_{n}) = 1, f_{2}(g_{1}, g_{2}, \ldots, g_{n}) = 1 \ldots, f_{n}(g_{1}, g_{2}, \ldots, g_{n}) = 1 $, then under what condition(s) we can separate $g_{1}, g_{2}, \ldots, g_{n}$ from the relations $f_{1}, f_{2}, \ldots, f_{n}$?
You appear to be asking whether, from the presentation of $G$, you can determine the orders of the generators. The answer is 'no'. In general, it is an insoluble problem to determine whether a group given by generators and relations is the trivial group.