Shorten broken branches back to an outward facing fork and cut there just above the fork. Large branches may need to be cut at the trunk. Prune just outside the branch collar, which is the raised area on the trunk where the branch meets the trunk. Cut branches on the underside first to avoid rip cuts down the trunk as the limb drops. Fallen trees are a mixed bag.
Rare specimens are worth trying to save.
I have seen trees that were replanted and they fall over again with every strong storm. This applies to palms as well as regular trees. A hurricane does provide an opportunity to plant superior trees to replace the fallen ones. Trees such as sapodilla, live oak, Indian tamarind, gumbo limbo and others have superior wind resistance and should be considered for future planting. Avoid weak sisters like ficus, ear leaf acacia, black olive and bischofia.