A group $G$ is locally solvable if all finitely generated subgroups are solvable.

A group $G$ is locally finite if all finitely generated subgroups are finite.

A group $G$ is virtually locally solvable if it has a locally solvable subgroup of the finite index.

Let be $R(S)=\left\langle T\,;\,T\trianglelefteq G\,,\,T\text{ locally solvable }\right\rangle$

My question are:

1)Is $\,R(S)\,$ locally solvable?

2) If 1) is true: $G$ locally finite, $R(S)$ locally solvable and $G/R(S)$ virtually locally solvable $\Rightarrow G$ virtually locally solvable?

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No, your group $R(S)$ need not be locally solvable. A counterexample due to P. Hall is described in Part 2 of Robinson's "Finiteness Conditions and Generalized Soluble Groups". See the development leading up to, and the proof of, Theorem 8.19.1 on page 91 (and the corollary).