We have an established logging system for our server-side services. Specifically, our Django project makes heavy use of the Python logging module, so call calls to logger.
First, I wrote and maintain log4javascript, so I'm declaring my interest up front. I also use it every day in my work, so I have some experience of it as a user. Here's how I would deal with your questions, specifically relating to log4javascript:
Use log4javascript's AjaxAppender for server logging;
debug, info, warning and error are all supported, as well as trace and fatal;
Use a BrowserConsoleAppender to log to FireBug or the native browser console;
If you don't want to remove all debug logging calls from you production code, you can either adjust your logger's threshold (using log.setLevel(log4javascript.Level.ERROR), for example, which will suppress all log calls with priority less than ERROR). If you want to suppress all logging calls, you can drop in a stub version of log4javascript in your production code.
You'll need to write a bit of code to do this using window.onerror. Something like window.onerror = function(msg, file, line) { log.error("Error in " + file + " on line " + line + ": " + msg); }
I'm not sure how you want to tie in with Google Analytics. log4javascript has no particular support for it.