About capacitors, C1 (smoothing capacitor) used to filter out twice the input frequency generated from D1 (full wave rectifier). Generally calculated based on how much ripple you would want to allow. Rule of thumb less than 10% of output voltage is preferred. This article tell you how to calculate.
Other capacitors C2 - C6 serves same purpose to decouple AC and provide stable DC output. Those values are often suggested in the datasheet.
For 78XX series IC product datasheet says the following,
The diode simply blocks DC Voltage. Diode D3 serves no purpose as even without it conducts only in forward direction. D2 and D4 in series treated as single diode as it doesn't enhance or affect any electrical properties except when forward biased will drop twice compared to single diode. Again in this schematic it serves no purpose at that location.