Those small pieces are called power pads, mainly found on older rackets. You can re-use them, or not, it's personal preference.
The knots with the piece of string in them are to keep the knots from pulling through the grommet. If you plan on tying off with a Parnell knot, you don't need it. Again it's personal preference if you do the same kind of knot.
Machines: Gamma 6004 2-point w/ Wise 2086 & Babolat Sensor Dual