Partial Resolution?

But there's still no: include ActionController::Layout::ClassMethods.

Nope. But in action_controller/layout.rb there is:
module ActionController #:nodoc:
  module Layout #:nodoc:
    def self.included(base)
      base.extend(ClassMethods)
      [...snip...]
    end
  end
end

HOW DO YOU KNOW, BOTTOM LINE, AFTER ALL THE VARIOUS FILES ARE
CONSIDERED, WHAT A CLASS DOES?

A few choices:
1) At runtime, ask it:
C:\...\site>ruby script\console
Loading development environment.

puts ActionController::Base.methods.sort

<
<=
<=>

···

From: gsm.beamon@gmail.com [mailto:gsm.beamon@gmail.com]

===
=~

=

__id__
[...snip...]
layout
layout_conditions
load
[...snip...]

2) Trust the author(s) of the code to write sufficient documentation
that explains to you all you need to know.

3) Read the source code.

I realize that none of these are ideal for huge projects like rails. I'm
not claiming that they are, I'm just laying out some options.

I still don't know how ApplicationController::Base can call
layout(). I still can't understand why it's not in the doc as common
sense demands.

Hopefully you now understand why it can call layout. To recap:

ActionController::Base.class_eval do
  include ActionController::Layout
end
...
module ActionController::Layout
  def self.included(base)
    base.extend(ClassMethods)
  end
end

Why it's not in the documentation is because:
1) RDoc was not smart enough to figure out that the end result of all
those runtime calls would always include ClassMethods methods as methods
of the ActionController::Base class.
and
2) Nobody writing the documentation for rails noticed this omission and
wrote additional explanatory code letting you know that all the methods
from ActionController::Layout::ClassMethods are available inside
ActionController::Base.