Does my mother's grandson have any legal rights to her house or claim anything if he has been in her house living off and on for at least 13 years?

Since  your mother has a will, and assuming it is valid, your nephew can't inherit anything through intestacy.  Since he is living in your mom's house with her permission, he has not gained any rights through adverse possession.  You don't mention any lease or other agreement in which your mother gave him permission to stay in the house after her death.  Thus, he would have no rights to the house.  However, anyone can claim anything they want, and you can't stop him from making a claim, however bogus.  Moreover, while he has no right to destroy any of your mother's property, that doesn't mean he won't do it.  if you feel it likely that he will destroy her property, you might have a better chance of safeguarding it if he was evicted, but that doesn't seem practical under the circumstances.  Theoretically, if you believe that your mother is no longer competent, you could ask a court to appoint you as her guardian which, if you were successful, would give you the legal power to evict your nephew, but besides the problem of enforcing the eviction, this coure of action would be expensive and would put you at odds with your mother, so I don't recommend it.  Ultimately this is not a legal quesiton, but a  practical one of how best to deal with your nephew, which only you can decide.