Why use capacitors with crystals?
You are actually building an oscillator circuit, using a crystal, some capacitors, and the internal circuitry of your microcontroller.  If you don't use the caps, your clocking won't work.  The values are calculated based on the properties of the crystal, as described in http://ww1.microchip.com/downloads/en/appnotes/00826a.pdf, page 9

Essentially, the load capacitance of the crystal, \$C_L\$, is set equal to the combination of the capacitors you're adding and the shunt capacitance of the crystal, so

$$ C_L = \frac{C^2}{2C} + C_{shunt}
$$

The ppm figures specify the (in)accuracy of the crystal (when properly used): 5 ppm is 5 parts per million.

Second and third overtones involve driving the crystal into resonance at a harmonic.  I suggest not trying this.