How to create a shared folder usable by my android app?

I've got a question that probably borders on opinion, but I've not any related questions or documentation that answers, so I feel like it's a fair one to ask. I'm trying to build an android app which modifies music files, and what I'd like to do is have a shared folder so that the files and the results can be accessible and shared. I'd like it if it was among the other folders like Music, Downloads, Movies, etc, or even under Music since it's music related. However this seems like it's a security no no in Android, as after I've made something and put it in there I have to use an intent to access it again, where as I'd rather just be able to open the files and not have a permissions based fiasco. Maybe some type of symbolic link like in Linux that pointed to my apps internal folder could be used, but of this I'm still uncertain. In any case, is there a way I should go about this? If so, are there some resources I could be pointed to?

Thank you in advance to anyone who takes this up!

Edit for CommonsWare:

I used the following to create the folder: File mediaStorageDir = new File(Environment.getExternalStorageDirectory(), APP_NAME);

And this to copy files from elsewhere to there:

public void copyFileToHomeDirectory(Uri uri)
{
    try
    {
        ContentResolver contentResolver = getApplicationContext().getContentResolver();
        String fileName = queryName(contentResolver, uri);
        //Get file extension
        String fileType = fileName.substring(fileName.length() - 4, fileName.length());
        if(fileType.equalsIgnoreCase(MP3_EXTENSION))
        {
            String path = Environment.getExternalStorageDirectory() + APP_FOLDER;
            InputStream in = contentResolver.openInputStream(uri);
            File outputFile = new File(path + File.separator + fileName);
            outputFile.createNewFile();
            OutputStream out = new FileOutputStream(outputFile);

            //First we crack open the file to copy it's contents:
            byte[] buffer = new byte[KB_SIZE];
            int read;
            while ((read = in.read(buffer)) != -1)
            {
                out.write(buffer, 0, read);
            }
            in.close();
            in = null;

            // write the output file (You have now copied the file)
            out.flush();
            out.close();
            out = null;
        }
    }
    catch(FileNotFoundException fnfe)
    {
        Log.e(TAG, "FileNotFoundException");
        Log.e(TAG, Log.getStackTraceString(fnfe));
    }
    catch(IOException ioe)
    {
        Log.e(TAG, "IOException");
        Log.e(TAG, Log.getStackTraceString(ioe));
    }
    catch(Exception e)
    {
        Log.e(TAG, "General Exception");
        Log.e(TAG, Log.getStackTraceString(e));
    }
}

I've tried other methods that I've overwritten in the process, but accessing the files to be used again I need something like this:

public void openDirectory(View view)
{
    // Choose a directory using the system's file picker.
    Intent intent = new Intent(Intent.ACTION_GET_CONTENT);

    // Provide read access to files and sub-directories in the user-selected
    // directory.
    //intent.addFlags(Intent.FLAG_GRANT_READ_URI_PERMISSION);
    //intent.putExtra(Intent.EXTRA_ALLOW_MULTIPLE, true);
    //intent.addCategory(Intent.CATEGORY_OPENABLE);

    // Optionally, specify a URI for the directory that should be opened in
    // the system file picker when it loads.
    //intent.putExtra(DocumentsContract.EXTRA_INITIAL_URI, uriToLoad);

    intent.putExtra(Intent.EXTRA_ALLOW_MULTIPLE, true);
    intent.addCategory(Intent.CATEGORY_OPENABLE);
    intent.setType("*/*");  //use image/* for photos, etc.

    //The result of this code will be calling the onActivityResult function below
    startActivityForResult(intent, REQUEST_MUSIC_DIR);
}

Edit2: I've reorganized the folders to what I think I should be doing so that I can work with the files freely, however, even in my internal cache storage (getCacheDir() + folder_name) either isn't letting me create the files (outputFile.createNewFile doesn't throw an error) or it isn't letting me open them when I go to get a directory listing.

Here's my code for creating the file:

String path = getCacheDir() + MY_SUB_FOLDER;

//uri is obtained through ACTION_OPEN_DOCUMENT intent
InputStream in = contentResolver.openInputStream(uri);
File outputFile = new File(path + "/" +  fileName);
outputFile.createNewFile();
Log.i(TAG, "The new file's directory/path is: " + outputFile.getAbsolutePath());

//NOTE: This is returning /data/user/0/com.example.myapplication/cache/MY_SUB_FOLDER/file_name.mp3

OutputStream out = new FileOutputStream(outputFile);

byte[] buffer = new byte[1024];
int read;
while ((read = in.read(buffer)) != -1)
{
   out.write(buffer, 0, read);
}
in.close();
in = null;

out.flush();
out.close();
out = null;

This is my code for attempting to open and read these newly created files

        File directory = new File(getCacheDir(), MY_SUB_FOLDER);
        Log.i(TAG, "This is the directory we're trying to get the files from: " + directory.getAbsolutePath());

//NOTE: This returns /data/user/0/com.example.myapplication/cache/MY_SUB_FOLDER
        File[] files = directory.listFiles();
        if(files != null)
        {
            for(int i = 0; i < files.length; i++)
            {
               Log.d(TAG, "Files found: " + files[i].getAbsolutePath());
            }
        }

The files variable isn't null but it's length is 0 and no files are found.

Edit3: I am catching the exceptions and logging any stack traces, which currently returns nothing.

        catch(FileNotFoundException fnfe)
        {
            Log.i(TAG, "FileNotFoundException");
            Log.i(TAG, Log.getStackTraceString(fnfe));
        }
        catch(IOException ioe)
        {
            Log.i(TAG, "IOException");
            Log.i(TAG, Log.getStackTraceString(ioe));
        }
        catch(Exception e)
        {
            Log.i(TAG, "General Exception");
            Log.i(TAG, Log.getStackTraceString(e));
        }