I found this stackoverflow post, which leads to the documentation on the run function of a GtkDialog, where it states:

The run() method blocks in a recursive main loop until the dialog
either emits the "response" signal, or is destroyed. If the dialog is
destroyed, the run() method returns gtk.RESPONSE_NONE; otherwise, it
returns the response ID from the "response" signal emission. Before
entering the recursive main loop, the run() method calls the
gtk.Widget.show() on the dialog for you. Note that you still need to
show any children of the dialog yourself.

During the run() method, the default behavior of "delete_event" is
disabled; if the dialog receives a "delete_event", it will not be
destroyed as windows usually are, and the run() method will return
gtk.RESPONSE_DELETE_EVENT. Also, during the run() method the dialog
will be modal. You can force the run() method to return at any time by
calling response() to emit the "response" signal. Destroying the
dialog during the run() method is a very bad idea, because your
post-run code won't know whether the dialog was destroyed or not.

After the run() method returns, you are responsible for hiding or
destroying the dialog as needed.

And also from the PyGTK 3 tutorial:

Finally, there are two ways to remove a dialog. The Gtk.Widget.hide()
method removes the dialog from view, however keeps it stored in
memory. This is useful to prevent having to construct the dialog again
if it needs to be accessed at a later time. Alternatively, the
Gtk.Widget.destroy() method can be used to delete the dialog from
memory once it is no longer needed. It should be noted that if the
dialog needs to be accessed after it has been destroyed, it will need
to be constructed again otherwise the dialog window will be empty.

So following on from those bits of information, if you call the hide function of the dialog before it receives the delete event, it will not be destroyed and you can keep calling run, to bring the dialog to focus.

e.g.

def on_mnu_test_dialog_activate(self, widget, data=None):

    result = self.TestDialog.run()

    if result == Gtk.ResponseType.OK:

        print "The OK button was clicked"

    self.TestDialog.hide()

Also, just to answer your second question.

In my example I have imported the class for the dialog:

from dialog_show.TestDialog import TestDialog

Then in the finish_initializing function, I created an instance variable for the dialog:

self.TestDialog = TestDialog()

I can then access a properties like so:

self.TestDialog.ui.txt_entry1.get_text()

or, as suggested by John,

self.TestDialog.builder.get_object("txt_entry1").get_text()