How much energy do islands spend compared to the rest of the world? Of course there will be some places where fossil fuels will be the only alternative. But using that as an excuse for inaction in the parts where we do have alternatives does not make sense.
Only with huge amounts of storage. So far there are no large, industrialized areas which have managed to build enough storage to maintain base load through multiple days of little sunshine or wind. It's a nice idea in theory but costs will have to come way down before we can afford to deploy storage at scale (especially in areas where the geography makes pumped hydroelectric impractical).
Currently we're talking about small islands. But why would an industrialized area built enough storage now if we don't have enough supply? Germany for example gets about 40-50% of its electricity from renewables (averaged over a year), with hardly any storage. Yet only a few percent of the renewable supply are wasted. We're very far away from getting 50% of our primary energy from renewables, so we could plausibly wait with building storage until that goal is achieved. If we had a proper carbon price, we could let the market figure out when it is most economical to start building storage for multiple days...