I think the authority is by default at the local level ie the incident commander, at least that is what we were taught. If something happened where I work the supervisor in charge becomes the incident commander and has the power to call in all available resources up to and including the federal level but the authority remains with him, unless he turns it over. The local incident commander is boss over whatever local, state, or federal help comes into the picture. So the authority belongs to the mayor and his local incident commanders. The authority isn't something the feds "own" and can "give to" the mayor, he and his fire/police chiefs already have it. He and his fire/police/etc. chiefs must not have a clear understanding how disaster management works.
I don't think so.
The only thing this Mayor knows is which end of the pipe You put in Your mouth and which end You light.