Is there a listener to listen when IDEA starts ?

For opening the projects, you can implement the

com.intellij.openapi.startup.StartupActivity.DumbAware#runActivity

and add a

<postStartupActivity id="<provide your ID>" implementation="<the class that implements com.intellij.openapi.startup.StartupActivity.DumbAware>"/>

inside the <extensions> on the META-INF/plugin.xml

but if you want to execute something after loading the project and after the indexing finishes, you can implement the

com.intellij.openapi.project.ProjectManagerListener#projectOpened

and add it inside the <projectListeners>

<listener class="<class that implements ProjectManagerListener>"
          topic="com.intellij.openapi.project.ProjectManagerListener"/>

of META-INF/plugin.xml

In addition, see https://www.jetbrains.org/intellij/sdk/docs/basics/plugin_structure/plugin_components.html