1. DO I need a application server to deploy a web service ?
No. Most WS implementations come in the shape of a web app, so what you need is a servlet container.
2. what do I need to create a web service ? Is Axis necessary tool ? Is it true that from JDK 6 already contains web service package so Axis is no longer needed ?
Axis is just one implementation, so it is not required - any implementation will do. Which implementation you can use depends on the kind of service you want to create. Jersey is a possibility for REST services (which are based on the JAX-RS API), and both Axis and the reference implementation (RI) are good choices for SOAP services that are based on the JAX-WS API.
JDK 6 does come with the JAX-WS RI, but I usually advise against using that - it's generally a couple of versions behind. See Deploy JAXWS (Mustang) WS to Tomcat? on how to get the RI up and running in Tomcat.
3. Is J2ee required to create web service ?
Ping & DNS - updated with new look and Ping home screen widget