http.server: Factorize request handling even more

This splits even more parts of the request handling code into a separate
class. Doing so allows us to reuse this part of the code for tests (e.g.
mocks). Unlike before now the error handling can also be reused.

The patch became a bit more convoluted than intended, but the end result
is easier to read than the original code.

Signed-off-by: Michael Hanselmann <hansmi@google.com>
Reviewed-by: René Nussbaumer <rn@google.com>