End user experience monitoring tools
There are some freely available tools for performance monitoring. Yahoo open-sourced a script they used called Boomerang which can measure page load times and other performance metrics for end-users. Full documentation here. Google analytics also offers a basic page load time report.
For error monitoring, you'll want to listen for the window.onerror event. I don't know of any scripts that will automatically log it for you, or mine the logs on the server side. If you implement your own, you'll want to be very careful about not pinging the server too often--imagine how many requests it would generate if there was a JS error in your JS error handling code!
Bucky Client and Bucky Server, can perform that task :
http://github.hubspot.com/bucky/
From their website :
Open-source tool to measure the performance of your web app directly from your users' browsers.
To analyse data they advise Graphite or OpenTSDB
You can try Atatus which provides Real User Monitoring(RUM) and Advanced error tracking for websites and web apps.