mkdir() works while inside internal flash storage, but not SD card?

First, you should note that file.mkdir() and file.mkdirs() returns false if the directory already existed. If you want to know whether the directory exists on return, either use (file.mkdir() || file.isDirectory()) or simply ignore the return value and call file.isDirectory() (see the documentation).

That said, your real problem is that you need permission to create the directory on removable storage on Android 5.0+. Working with removable SD cards on Android is horrendous.

On Android 4.4 (KitKat), Google restricted access to SD cards (see here, here, and here). See this StackOverflow answer which leads to this XDA post if you need to create a directory on a removable SD card on Android 4.4 (KitKat).

On Android 5.0 (Lollipop), Google introduced new SD card access APIs. For sample usage please refer to this stackoverflow answer.

Basically, you need to use DocumentFile#createDirectory(String displayName) to create your directory. You will need to ask the user to grant permissions to your app before creating this directory.

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NOTE: This is for removable storage. Using File#mkdirs() will work on internal storage (which is often confused with external storage on Android) if you have the permission android.permission.WRITE_EXTERNAL_STORAGE.

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I will post some example code below:

Check if you need to ask for permission:

File sdcard = ... // the removable SD card
List<UriPermission> permissions = context.getContentResolver().getPersistedUriPermissions();
DocumentFile documentFile = null;
boolean needPermissions = true;

for (UriPermission permission : permissions) {
  if (permission.isWritePermission()) {
    documentFile = DocumentFile.fromTreeUri(context, permission.getUri());
    if (documentFile != null) {
      if (documentFile.lastModified() == sdcard.lastModified()) {
        needPermissions = false;
        break;
      }
    }
  }
}

Next (if needPermissions is true), you can display a dialog to explain to the user that they need to select the “SD Card” to give your app permissions to create files/directories and then start the following activity:

if (needPermissions) {
  // show a dialog explaining that you need permission to create the directory
  // here, we will just launch to chooser (what you need to do after showing the dialog)
  startActivityForResult(new Intent(Intent.ACTION_OPEN_DOCUMENT_TREE), STORAGE_REQUEST_CODE);
} else {
  // we already have permission to write to the removable SD card
  // use DocumentFile#createDirectory
}

You will now need to check the resultCode and requestCode in onActivityResult:

@Override protected void onActivityResult(int requestCode, int resultCode, Intent data) {
  if (requestCode == STORAGE_REQUEST_CODE && resultCode == RESULT_OK) {
    File sdcard = ... // get the removable SD card

    boolean needPermissions = true;
    DocumentFile documentFile = DocumentFile.fromTreeUri(MainActivity.this, data.getData());
    if (documentFile != null) {
      if (documentFile.lastModified() == sdcard.lastModified()) {
        needPermissions = false;
      }
    }

    if (needPermissions) {
      // The user didn't select the "SD Card".
      // You should try the process over again or do something else.
    } else {
      // remember this permission grant so we don't need to ask again.
      getContentResolver().takePersistableUriPermission(data.getData(),
          Intent.FLAG_GRANT_READ_URI_PERMISSION | Intent.FLAG_GRANT_WRITE_URI_PERMISSION);
      // Now we can work with DocumentFile and create our directory
      DocumentFile doc = DocumentFile.fromTreeUri(this, data.getData());
      // do stuff...
    }
    return;
  }
  super.onActivityResult(requestCode, resultCode, data);
}

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That should give you a good start on working with DocumentFile and removable SD cards on Android 5.0+. It can be a PITA.

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Also, there is no public API to get the path to a removable SD card (if one even exists). You should not rely on hardcoding "/storage/sdcard1"! There are quite a few posts about it on StackOverflow. Many of the solutions use the environment variable SECONDARY_STORAGE. Below is two methods you can use to find removable storage devices:

public static List<File> getRemovabeStorages(Context context) throws Exception {
  List<File> storages = new ArrayList<>();

  Method getService = Class.forName("android.os.ServiceManager")
      .getDeclaredMethod("getService", String.class);
  if (!getService.isAccessible()) getService.setAccessible(true);
  IBinder service = (IBinder) getService.invoke(null, "mount");

  Method asInterface = Class.forName("android.os.storage.IMountService$Stub")
      .getDeclaredMethod("asInterface", IBinder.class);
  if (!asInterface.isAccessible()) asInterface.setAccessible(true);
  Object mountService = asInterface.invoke(null, service);

  Object[] storageVolumes;
  if (Build.VERSION.SDK_INT >= Build.VERSION_CODES.M) {
    String packageName = context.getPackageName();
    int uid = context.getPackageManager().getPackageInfo(packageName, 0).applicationInfo.uid;
    Method getVolumeList = mountService.getClass().getDeclaredMethod(
        "getVolumeList", int.class, String.class, int.class);
    if (!getVolumeList.isAccessible()) getVolumeList.setAccessible(true);
    storageVolumes = (Object[]) getVolumeList.invoke(mountService, uid, packageName, 0);
  } else {
    Method getVolumeList = mountService.getClass().getDeclaredMethod("getVolumeList");
    if (!getVolumeList.isAccessible()) getVolumeList.setAccessible(true);
    storageVolumes = (Object[]) getVolumeList.invoke(mountService, (Object[]) null);
  }

  for (Object storageVolume : storageVolumes) {
    Class<?> cls = storageVolume.getClass();
    Method isRemovable = cls.getDeclaredMethod("isRemovable");
    if (!isRemovable.isAccessible()) isRemovable.setAccessible(true);
    if ((boolean) isRemovable.invoke(storageVolume, (Object[]) null)) {
      Method getState = cls.getDeclaredMethod("getState");
      if (!getState.isAccessible()) getState.setAccessible(true);
      String state = (String) getState.invoke(storageVolume, (Object[]) null);
      if (state.equals("mounted")) {
        Method getPath = cls.getDeclaredMethod("getPath");
        if (!getPath.isAccessible()) getPath.setAccessible(true);
        String path = (String) getPath.invoke(storageVolume, (Object[]) null);
        storages.add(new File(path));
      }
    }
  }

  return storages;
}

public static File getRemovabeStorageDir(Context context) {
  try {
    List<File> storages = getRemovabeStorages(context);
    if (!storages.isEmpty()) {
      return storages.get(0);
    }
  } catch (Exception ignored) {
  }
  final String SECONDARY_STORAGE = System.getenv("SECONDARY_STORAGE");
  if (SECONDARY_STORAGE != null) {
    return new File(SECONDARY_STORAGE.split(":")[0]);
  }
  return null;
}