Is there a web app for Emacs-Lisp API searching with nice description, links, examples, etc?
Something like http://apidock.com/ for Ruby, or http://en.cppreference.com/w/ for C++, etc.
No, there isn't anything as nice as those online docs.
That being said, your Emacs is the best source of information. The online Emacs reference only applies to the latest released version of Emacs, so some of it may not apply to your version.
I highly recommend the helm-apropos command from the Helm package, which allows for incremental completion of function, command, variable, and face names. I find it invaluable for exploring a new Emacs package.
Also consider the apropos-documentation command, which allows you to search inside docstrings.
As for external documentation, take a look at Artur Malabarba's Emacs Online Documentation (also available as an Android app!).
M-x apropos,C-h fand similar aids).C-h i,C-h fandC-h Sare your best friends. Also consider asking on the mailing list.