How/When to generate Gradle wrapper files?

I am trying to understand how the Gradle Wrapper works. In many source repos, I see the following structure:

projectRoot/
    src/
    build.gradle
    gradle.properties
    settings.gradle
    gradlew
    gradlew.bat
    gradle/
        wrapper/
            gradle-wrapper.jar
            gradle-wrapper.properties

My questions:

1. How/when does one generate gradlew/gradlew.bat? Are you supposed to generate them only one time when the project is first created, do you generate them every time you commit/push changes? And how are they generated?
2. Same question above, but for the gradle/wrapper/* files (gradle-wrapper.jar and gradle-wrapper.properties)?
3. Some times I see other *.gradle files inside the project's gradle directory. What are these additional Gradle files and what do they represent/do? Custom plugins?
4. What is the difference in properties that go into settings.gradle vs what should be defined inside gradle.properties?

1. You generate it once, and again when you'd like to change the version of Gradle you use in the project. There's no need to generate is so often. Here are the docs. Just add wrapper task to build.gradle file and run this task to get the wrapper structure.

   Mind that you need to have Gradle installed to generate a wrapper. Great tool for managing g-ecosystem artifacts is SDKMAN!. To generate a gradle wrapper, add the following piece of code to build.gradle file:

   task wrapper(type: Wrapper) {
      gradleVersion = '2.0' //version required
   }
   

   and run:

   gradle wrapper
   

   task. Add the resulting files to SCM (e.g. git) and from now all developers will have the same version of Gradle when using Gradle Wrapper.

   With Gradle 2.4 (or higher) you can set up a wrapper without adding a dedicated task:

   gradle wrapper --gradle-version 2.3
   

   or

   gradle wrapper --gradle-distribution-url https://myEnterpriseRepository:7070/gradle/distributions/gradle-2.3-bin.zip
   

   All the details can be found here

From Gradle 3.1 --distribution-type option can be also used. The options are binary and all and bin. all additionally contains source code and documentation. all is also better when IDE is used, so the editor works better. Drawback is the build may last longer (need to download more data, pointless on CI server) and it will take more space.

2. These are Gradle Wrapper files. You need to generate them once (for a particular version) and add to version control. If you need to change the version of Gradle Wrapper, change the version in build.gradle see (1.) and regenerate the files.

3. Give a detailed example. Such file may have multiple purposes: multi-module project, responsibility separation, slightly modified script, etc.

4. settings.gradle is responsible rather for structure of the project (modules, names, etc), while, gradle.properties is used for project's and Gradle's external details (version, command line arguments -XX, properties etc.)