I've come across the following example: $$ρ^3·σρ^2 = ρ^2σρ^{−1}ρ^2 = ρ^2σρ = ρσρ^{-1}ρ = ρσ = σρ^{−1} = σρ^5$$
And was wondering if it is true in general that $ρ^i·σρ^j = σρ^{i+j}$?
I know that $ρσ = σρ^{n-1}$ for the dihedral group $D_n$ so maybe that would help some how?
No, but what is true in general is $\rho^m \sigma \rho^k = \sigma \rho^{k-m}$.
This happens to correspond to $\rho^{k+m}$ in your specific example because (it appears you are working in $D_6$) $\rho^3=\rho^{-3}$.