Here is my recommendations for you.
1. What is the best IDE for developing Prime Faces applications?
Eclipse Indigo or IntelliJ. Choice is a matter of preference.
In Eclipse you can add the JSF 2.0 project Facet and you'll have code completion for component libraries, including PrimeFaces.
2. What are supported web servers to deploy and run Prime Faces applications?
I would receommend JBoss AS 7. It is fast, modular, incredible easy to use and supports hot deployment. Little or none configuration is required to run and deploy your apps.http://www.jboss.org/as7
I would recommend you to check out Seam Solder 3 and CDI/Weld. That's a fantastic way to build and get started. Seam Solder and -Faces replaces the Managed Bean facility by CDI Beans, in which provides a by far more powerful and simplified programming model. You'll have all what's in ManagedBeans + much more.http://seamframework.org/Seam3
It all depends on your requirements though.
PrimeFaces 2.x and 3 is fully compatible with Seam 3.
3. What are supported application servers to deploy and run Prime Faces applications?
GlassFish 3.x, JBoss 6 and 7, Tomcat 6+, Jetty and Eclipse Jetty. Servlet 2.5 is enough, but 3.0 is preferred.
4. Does it support Internet Explorer,Good Chrome,Mozilla Browsers?
Yes, it has good cross browser support. However, there are several browser compatibility issues left in current development. This is something the whole community should test and report on so it'll be fixed.