How to uniquely identify a control in a child form when multiple instances of the form exist

Question 1

You've one instance of BackgroundWorker and each ListForm you create, is registered to this worker. So you've to pass the instance of the Form to the worker.

Create a little Helper class with two Attributes. This is just an example. You could also pass an identifier or what ever you like:

public struct ReadAllArguments
{
    public bool Read;
    public ListForm CallingForm;

    public ReadAllArguments(bool read, ListForm callingForm)
    {
        Read = read; CallingForm = callingForm;
    }
}

You could pass it then like this:

...
    if (bw.IsBusy != true)
        {
            // Start the ReadAll parameters thread
            btnReadAll.Text = "Cancel Read";
            btnWriteAll.Enabled = false;
            bw.RunWorkerAsync(new ReadAllArguments(true, this));
        }
...

An later read it like that:

private void bw_DoWork(object sender, DoWorkEventArgs e)
    {
        BackgroundWorker worker = sender as BackgroundWorker;
        DoUIWorkHandler DoReadClick;
        DoUIWorkHandler DoWriteClick;

        int CurrentControlCount = 1;
        string StatusText = "";
        int ProgressValue = 0;
        ReadAllArguments arguments = e.Argument as ReadAllArguments;
        if (this != arguments.ListForm)
          return;

        ...

                        if (arguments.Read)
                        {
                            DoReadClick = c.btnRead.PerformClick;
                            c.Invoke(DoReadClick);
                        }
                        else
                        {
                            DoWriteClick = c.btnWrite.PerformClick;
                            c.Invoke(DoWriteClick);
                        }
       ...

You'll realize that you can even move the Work-Method out of you Form because there are no direct dependencies and you don't need access to the "this"-Qualifier. You've passed everything in you argument. After replacing every "this" by that argument you could register exactly one Work-Method to the DoWork-Event of your Worker. This would be much cleaner and more elegant...

Here's an example how you could do this:

public partial class ListForm: Form
{
    // Background Worker Thread for Read / Write All tasks
    private static BackgroundWorker bw = new BackgroundWorker();

    static ListForm()
    {
        //We move the do-work out of the instance constructor, because the work that has to be done, is not connected to our instances. So we've only one definition of our work that has to be done
        bw.DoWork += new DoWorkEventHandler(TheWorkThatHasToBeDone);
    }

    public ListForm()
        {
             InitializeComponent();

             // Configure the Background Worker that reads and writes all variable data
             bw.WorkerReportsProgress = true;
             bw.WorkerSupportsCancellation = true;
             //no more registering on instance level 
             bw.ProgressChanged += new ProgressChangedEventHandler(bw_ProgressChanged);
             bw.RunWorkerCompleted += new RunWorkerCompletedEventHandler(bw_RunWorkerCompleted);    
        }

    //Your new instance-independent doWork-Method - static here
    private static void TheWorkThatHasToBeDone(object sender, DoWorkEventArgs e)
    {
        BackgroundWorker worker = sender as BackgroundWorker;
        ReadAllArguments arguments = e.Argument as ReadAllArguments;
        //You call the instance-Method here for your specific instance you want the work to be done for
        arguments.ListForm.bw_DoWork(worker, arguments);
    }


    //Your old bw_DoWork-Method with nicer arguments - you should change the method name...
    private void bw_DoWork(BackgroundWorker worker, ReadAllArguments arguments)
        {
            DoUIWorkHandler DoReadClick;
            DoUIWorkHandler DoWriteClick;

            int CurrentControlCount = 1;
            string StatusText = "";
            int ProgressValue = 0;

            // *******************Perform a time consuming operation and report progress. 
            try
            {
                ...
            }
        }

It would again be more elegant to move the stuff out of the forms code and not doing this with static members, but I think the idea is clear.

Question 2

To identify object you could use HashCode or create an Id property, and next use it in custom EventArgs.

private Guid _controlId;

    public ListForm()
    {
        _controlId = Guid.NewGuid();
        ...
    }

Try also to menage the event observators in this way:

private void btnReadAll_Click(object sender, EventArgs e)
    {
        if (bw.IsBusy != true)
        {
            bw.DoWork += bw_DoWork;
            bw.ProgressChanged += bw_ProgressChanged);
            bw.RunWorkerCompleted +=bw_RunWorkerCompleted;

            // Start the ReadAll parameters thread
            btnReadAll.Text = "Cancel Read";
            btnWriteAll.Enabled = false;
            bw.RunWorkerAsync("R");
        }
        else if (bw.WorkerSupportsCancellation == true)
        {
            // Cancel the ReadAll parameters thread
            bw.CancelAsync();
        }

        bw.DoWork -= bw_DoWork;
        bw.ProgressChanged -= bw_ProgressChanged;
        bw.RunWorkerCompleted -= bw_RunWorkerCompleted;
    }