3. Should I consider AJAX?
AJAX definitely has the buzz right now, but it might not be the right thing for you. AJAX is limited to the latest browsers, exposes browser compatibility issues, and requires new skill-sets for many. There is a good blog entry by Alex Bosworth on AJAX Mistakes which is a good read before you jump full force into AJAX.
On the other hand you can achieve highly interactive rich web applications that are responsive and appear really fast. While it is debatable as to whether an AJAX based application is really faster, the user feels a sense of immediacy because they are given active feedback while data is exchanged in the background. If you are an early adopter and can handle the browser compatibility issues, and are willing to learn some more skills, then AJAX is for you. It may be prudent to start off AJAX-ifying a small portion or component of your application first. We all love technology, but just remember the purpose of AJAX is to enhance your user's experience and not hinder it.

4. Does AJAX work with Java?
Absolutely. Java is a great fit for AJAX! You can use Java Enterprise Edition servers to generate AJAX client pages and to serve incoming AJAX requests, manage server side state for AJAX clients, and connect AJAX clients to your enterprise resources. The JavaServer Faces component model is a great fit for defining and using AJAX components.

5. Won't my server-side framework provide me with AJAX?
You may be benefiting from AJAX already. Many existing Java based frameworks already have some level of AJAX interactions and new frameworks and component libraries are being developed to provide better AJAX support. I won't list all the Java frameworks that use AJAX here, out of fear of missing someone, but you can find a good list at www.ajaxpatterns.org/Java_Ajax_Frameworks.
If you have not chosen a framework yet it is recommended you consider using JavaServer Faces or a JavaServer Faces based framework. JavaServer Faces components can be created and used to abstract many of the details of generating JavaScript, AJAX interactions, and DHTML processing and thus enable simple AJAX used by JSF application developer and as plug-ins in JSF compatible IDE's, such as Sun Java Studio Creator.

6. Where should I start?
Assuming the framework you are using does not suffice your use cases and you would like to develop your own AJAX components or functionality I suggest you start with the article Asynchronous JavaScript Technology and XML (AJAX) With Java 2 Platform, Enterprise Edition.
If you would like to see a very basic example that includes source code you can check out the tech tip Using AJAX with Java Technology. For a more complete list of AJAX resources the Blueprints AJAX home page.
Next, I would recommend spending some time investigating AJAX libraries and frameworks. If you choose to write your own AJAX clients-side script you are much better off not re-inventing the wheel.
AJAX in Action by Dave Crane and Eric Pascarello with Darren James is good resource. This book is helpful for the Java developer in that in contains an appendix for learning JavaScript for the Java developer.