I've been trying to attack this problem in many ways, but Couldn't figure out the right answer .
The question is, That a finitely generated subgroup of SO3 when all the elements are of finite order, can not be dense . This is a preceding question to proving that these subgroups are finite (so this statement is forbidden) .
It can be proven alternatively that this subgroup must be closed, but this is also a tough argument .
The two questions also can be treated in a different order. We can first solve the second question. Indeed, all finitely generated torsion subgroups of $SO(3)$ are finite, see here:
Finitely generated torsion subgroup of $SO(3,\mathbb{R})$ is finite
Hence such subgroups cannot be dense.