Basic AJAX Question:

Answer #1
There are many libraries/frameworks out there (and many more emerging) that will help abstract such things as all the nasty browser differences. Three good libraries are The Dojo Toolkit, Prototype, and DWR.

* The Dojo Toolkit contains APIs and widgets to support the development of rich web applications. Dojo contains an intelligent packaging system, UI effects, drag and drop APIs, widget APIs, event abstraction, client storage APIs, and AJAX interaction APIs. Dojo solves common usability issues such as support for dealing with the navigation such as the ability to detect the browser back button, the ability to support changes to the URL in the URL bar for bookmarking, and the ability to gracefully degrade when AJAX/JavaScript is not fully support on the client. Dojo is the Swiss Army Knife of JavaScript libraries. It provides the widest range of options in a single library and it does a very good job supporting new and older browsers.

Answer #2
* Prototype focuses on AJAX interactions including a JavaScript AJAX object that contains a few objects to do basic tasks such as make a request, update a portion of a document, insert content into a document, and update a portion of a document periodically. Prototype JavaScript library contains a set of JavaScript objects for representing AJAX requests and contains utility functions for accessing in page components and DOM manipulations. Script.aculo.us and Rico are built on top of Prototype and provide UI effects, support for drag and drop, and include common JavaScript centric widgets. If you are just looking to support AJAX interactions and a few basic tasks Prototype is great. If you are looking for UI effects Rico and Script.aculo.us are good options.
* Yahoo UI Library is a utility library and set of widgets using the APIs to support rich clients. The utility library includes support for cross-browser AJAX interactions, animation, DOM scripting support, drag and drop, and cross browser event support. The Yahoo UI Library is well documnented and contains many examples.

Answer #3
* DWR (Dynamic Web Remoting) is a client-side and server-side framework that focuses on allowing a developer to do RPC calls from client-side JavaScript to plain old Java objects in a Java Enterprise Edition web container. On the server side DWR uses a Servlet to interact with the Java objects and returns object representations of the Java objects or XML documents. DWR will be easy to get up and running and plays well with other Java technologies. If you are looking for a client-side and server-side framework that integrates well use DWR.
* Google Web Toolkit (GWT) is client/server framework provided by Google that allows a developer to write an AJAX application in pure Java. The GWT takes care of the details of generating all the client-side code using a Java-to-JavaScript compiler. One of the key benefits of the GWT Software Developer Kit (SDK) is that it allows you to debug your applications in what is known as GWT hosted mode using an embedded browser (IE on Windows and Mozilla/Gecko on Linux) that is tied to the toolkit. In GWT hosted mode you setup through the code and debug it as it is running on both the client and server. The GWT contains a default set of widgets and widget containers. An application is built by coding a set of widgets and containers together much like would be done in a Swing application.

Answer #4
The GWT Software Developer Kit (SDK) is limited to Linux and Windows XP/2000 though the web applications it generates are compatible with the latest generation of the mainstream browsers.

There are many new and emerging libraries for JavaScript and this list only reviews some of the more common libraries. When making a choice choose the library which suites your needs the best. While it might be better to choose one, there is nothing stopping you from using more than one framework. For a more extensive list of client-side frameworks see: Survey of AJAX/JavaScript Libraries.