Welcome to JavaRanch.
1) Axis is a webapp, is it mandatory to create my webservice within this webapp or can I just take a few library?
You can integrate Axis into your own web app. It's just a
servlet, after all (that happens to need a lot of supporting libraries).
2) Do I have to call my web service method within an url with the required parameters or can I just call the basic url and attach it a SOAP message containing the name of the method and the parameters?
I'm not quite sure what you're asking. Axis does not support REST-style WS (where parameters are transported in the URL). All methods, parameters and results are transported in the SOAP body of the HTTP request/response.
3) Is it a good thing to start with axis or could I choose another library?
Other WS toolkits are available, but Axis is widely used that you'll find it easier to get answer to your questions, and there are lots more articles and tutorials available for it than for other toolkits.
4) Does anyone has a simple example to share with me?
The
WebServicesFaq page links to several start-from-scratch-with-Axis articles (in the "How do I get started building a web service with Axis ?" section). It also links to a plethora of other resources you'll find useful while getting into WS.