What rights does the executor of a Will have?

A valid will is enforceable. A person named as executor in the will has power or authority over the estate. However, someone named in person A's will does NOT become the executor or personal representative of person B's estate, even if much or many of the assets of person B originally came from person A. So your grandfather's estate will have its own personal representative and will therefore--since he had no will--pass by the rules for intestate succession (who gets what when there is no will) without any reference to your grandmother's will.
The critial issue is, however, what actually was in your grandfather's estate? Obviously, anything he personally owned is in his estate. Anything he inherited from your grandmother is in it, too. But he may not have inherited "everything" legally. If your grandmother's will did not leave everything to him, he would NOT have automatically received it: no state requires you to leave *everything* to a spouse. He was guaranteed to get some of your grandmother's estate by RI's "elective share" law, which prohibits completely disinheriting a spouse, and certain jointly owned assets (money in a joint bank account; a house owned as joint tenants with right of survivorship) or "pay on death" (POD) or "transfer on death" (TOD) assets would go to him, but unless her will specifically left everything to him, it is very likely that other people (the other people named in your grandmother's will) would receive at least something from her--and anything received by other people should have been distributed according to her will, are not part of his estate, and should not be under the control of his estate's personal reprsesentative.
It may be possible, assuming that probate on your grandmother's estate was not completed, to challenge what happened to or with those assets she left to other people--again, there is no law that your grandfather would have automatically recieved everything, unless everything they owned was owned as a joint asset. This is a complicated situation--you will effectively need an attorney to help you sort it out. If you believe that the portion of your grandmother's estate which she would  have left to other people (i.e. you, if you are going to take on the effort and cost of this matter) is large enough to justify the cost of an attorney, then consult with a probate attorney. Otherwise, if the feel there is not enough money at stake to justify paying several thousand dollars (at least) to a lawyer, you may wish to simply leave this alone--disentangling two different estates, as this will entail, could be an expensive undertaking in court.