Going back to Kindle firmware 3.2.x, the experimental browser absolutely supports JavaScript (ES3 spec), some CSS 2.x, and scores 55 (our of 555 possible points) on HTML5Test.com. It more or less passes the Acid3 browser test at 100%. This puts it on significantly better footing than Internet Explorer 8, other than on raw JavaScript performance benchmarks.
Strictly speaking, it is not an HTML5-capable browser despite having a non-zero score on HTML5Test.com. It doesn't support any HTML5 document features, but at the same time supports relatively advanced features like Web Workers, Cross-Document messaging, and Cross-Origin Resource Sharing.
With our Kindle 2 with International 3G, we were able to check Yahoo email, Gmail, Wikipedia, and some Maps from a remote site in Taiwan while on vacation. You can jailbreak a Kindle 2 to install the Kindle 3.x firmware. Any Kindle after the Kindle 2 can be updated to the latest 3.x firmware and have a quite functional, albeit archaic, browser compared to competing e-ink devices.
Even the very latest Kindle e-ink devices (firmware 5.8.x) only score 152 (out of 555) on HTML5Test.com, on par with Internet Explorer 9 which was 2 years behind competing browsers when it was released 6 years ago. They support some aspects of the ES5.1 JavaScript standard, but several aspects are missing/broken. It has partial support for WebSockets that makes it unusable for most web apps that use that feature, but no support for Server Sent Events which is bizarre for a device where battery life is critical. Amazon continues their history of what appears to be a purposefully broken CSS2.1 and CSS3 implementation, and the browser will hang or crash when trying popular benchmark sites like JetStream, ARES- 6, or Ringmark. One cool saving grace is the inclusion of Local Storage and Canvas support, which would make it possible to have games with decent functionality if their animations are optimized for e-ink refresh rates. The Kindle browser doesn't support web standard touch events in the browser, but there's other control possibilities a developer could employ.
That being said, even Kindle firmware 5.8.x is a decent browser on a device that has weeks-long battery life. It will reasonably render the low-end mobile (read: iOS and Android 2.x) versions of Twitter, Facebook, Wikipedia, and other major sites with only minor render issues. Amazon can and should provide a better web experience given the prices they charge, but in the worst case scenario the jailbreak community compensates wonderfully on the software side.