And how do a bunch of lumbering unarmed dead people overwhelm an Abrams tank? Even assuming they could take out army bases on land, what about the navy? Where are the carrier battlegroups in the middle of the Pacific Ocean? Zombies couldn't touch them!
I'm hardly a "Zombie Apocalypse" fanatic (I watched a few of the Walking Dead shows...they were entertaining).
However - the questions that you asked were largely answered by a book called "World War Z". Look it up, I enjoyed it. It was a fast read.
If everyone gets inside a ship on port and goes out to sea, the zombies cannot reach them. Even getting into docked or beached solid ship in port and pulling up all the ladders works - they can’t climb the sides of ship.
A large quantity of people, especially people that feel no pain or fear, can over run any position. Eventually you run out of ammo, and fuel, and traction, and morale.
The Navy very well might still be out there. Problem is to be useful they need to land, which means they become a ground force, which runs into all those problems. Sure you can bomb the cities from the air, nice and safe from the zombies, but you’re also killing civilians, and burning up that finite supply of ammo and fuel and morale (people really tend to not like bombing their own cities).
Yes, there are plenty of people who would survive this kind of apocolypse, from everyone on the ocean, to the extremely rural, miners, etc. While I thought the first episode was well done, I didn't like that the protagonist woke up in a hospital. This has already been done in a zombie-ish movie, 28 days later. It would have been cool to see some miners get cut off from the surface, fight their way to the top, come out into a zombie apocolypse, and realize the people on the surface were protecting them by cutting them off.