> A second issue with \left(\right) is that typing "\left(" produces
> "\left(()\right)" instead of "\left(\right)".

Can't reproduce that either.  Typing "\left(" completes to
"\left(\right)" for me.

Can you please start Emacs with

    emacs -Q &

and in scratch buffer, eval these lines:

    (require 'package)
    (package-initialize)

open your .tex file and see if happens again?

Best, Arash