Finding an algorithm that minimizes vertex weight sum of a subgraph that satisfies several constraints
Without loss of generality we can assume the graph is connected.  (If not, decompose it into connected components, and solve the problem separately on each component.)  Since it is connected, every vertex has a path to every other, so we can ignore the requirement to have a path to a root.
So your problem becomes: given $R$, find a minimum dominating set $V'$ that contains $R$.  This is NP-complete; indeed, the special case where $R=\emptyset$ is the standard dominating set problem, which is NP-complete, so your problem is, too.
If you need to solve your problem in practice, you could apply any of the standard techniques for dominating set or set cover.  For instance, you could use an approximation algorithm or greedy heuristic to get an approximate solution; or you could use a SAT solver or ILP solver to get an exact solution and hope that it terminates in a reasonable amount of time on your specific problem instances.