As for #2, I suspect that the practicalities of needing room to periodically load/unload fuel rods from the top of the reactor vessel is why the control rods come up from the bottom.
Good point. It would make it more difficult from above. I'll be the first to admit I don't know much beyond the basics about nuclear reactors. It just seems strange that an electricity producer would have to have grid power to operate.
Under normal operating conditions, a nuclear power plant needs no external source of electricity. The cooling water pumps and other auxilary machinery are powered by using a small proportion of the output power of the reactor.
If however the reactor is shut down, then no power is produced, but the reactor core still produces heat for some time.
Under these conditions power must be either produced on site or imported from the grid, unless the reactor is designed for passive cooling.
The heat produced from a shut down reactor is not constant but steadily reduces, eventually to a level low enough for passive cooling.