How can I write a file to a folder of the internal storage on Android?

As you have figured out writing to root directories in Android is impossible unless you root the device. Thats why even some apps in Play-store asking for root permissions before installing the app. Rooting will void your warranty so i don't recommend it if you don't have serious requirement.

Other than root directories you can access any folder which are visible in your Android file manager.

Below is how you can write into sd with some data - Taken from : https://stackoverflow.com/a/8152217/830719

Use these code you can write a text file in SDCard along with you need to set permission in android manifest

<uses-permission android:name="android.permission.WRITE_EXTERNAL_STORAGE" />

this is the code :

public void generateNoteOnSD(String sFileName, String sBody){
    try
    {
        File root = new File(Environment.getExternalStorageDirectory(), "Notes");
        if (!root.exists()) {
            root.mkdirs();
        }
        File gpxfile = new File(root, sFileName);
        FileWriter writer = new FileWriter(gpxfile);
        writer.append(sBody);
        writer.flush();
        writer.close();
        Toast.makeText(this, "Saved", Toast.LENGTH_SHORT).show();
    }
    catch(IOException e)
    {
         e.printStackTrace();
         importError = e.getMessage();
         iError();
    }
   }  

.