well the cut/copy/paste is usually in the Control that is TextBox based...
this.theTextBox.Copy(); //you need to set the selectionstart/selectionlength properties so it can copy it from that.
in regards to shortcut keys - in what sense The standard Windows shortcut keys for cut/copy/paste still apply even for your app as they are standard Windows shortcut keys. Otherwise to make your own I guess you need to implement the keydown event of the control and check the KeyEventArgs for which key they pressed and perform your action based on that selection.
private void textbox1_keydown(object sender, KeyEventArgs e)
if (e.KeyData == Keys.H)
//user pressed the h key in the textbox1 control