More accurately, the Tomcat deployer reads, digests, and applies web.xml to build Tomcat-internal app support objects. The application itself doesn't access web.xml, but it's one of 2 deployment descriptors used when a webapp is deployed. Web.xml is the container-independent deployment spec. The other deployment descriptor is container-specific, and in the case of Tomcat, it's the Context definition supplied or synthesized when a webapp is deployed.
Under normal circumstances, all eligible webapps are deployed at server startup and when the webapp WAR or Context definition is changed. Redeploying on a change is optional, but enabled by default. Tomcat scans the webapps at intervals and redeploys eligible webapps, so the redeployment isn't quite instantaneous, but it's a fairly short interval by default.
An IDE is no substitute for an Intelligent Developer.