At a minimum, the 5 VDC power supply must source this amount of current:
I_OUT = Arduino Nano nominal = 19 mA
+ Arduino Nano D2 output current = current into 2N2222 base
+ Arduino Nano D6 output current = current BS170 gate
+ Relay coil = 79.4 mA
+ Buzzer = 70 mA
As a best practice, design the 5 VDC power supply to handle at least 30% more current than the nominal/typical operating current that's drawn from the supply by the project.
Let's assume you've chosen \$\beta_{sat}=10\$ for the 2N2222 NPN transistor. Therefore, current that must be sourced from the Nano's D2 pin into the NPN transistor's base is:
$$
I_{B_{sat}} = \frac{I_{C_{sat}}}{\beta_{sat}} = \frac{I_{C_{sat}}}{10}
$$
where \$I_{C_{sat}}\$ is the transistor's collector current when the transistor is operating in saturation mode (fully "ON").
The value for the NPN transistor's base current limiting resistor is calculated as follows:
$$
R_B = \frac{V_{R_B}}{I_{RB}} = \frac{V_{OH}-V_{BE_{sat}}}{I_{B_{sat}}}
$$
where \$V_{OH}\$ is the minimum voltage for a logic HIGH output (~4.2 VDC when the DIO pin is sourcing 20 mA), and \$V_{BE_{sat}}\$ is the NPN's base-emitter voltage when operating in saturation mode with a collector current of \$I_{C_{sat}}\$. (Example: If the optocoupler LED current is 50 mA when ON, then \$I_{C_{sat}}=50\,mA\$, \$I_{B_{sat}}=I_{C_{sat}}/\beta_{sat}=50\,mA / 10 = 5\,mA\$, and \$V_{BE_{sat}}\;(@I_{C_{sat}}=50\,mA)\;\approx 0.7\,VDC\$.)
Getting back to the 5 VDC power supply, the power it dissipates is approximately:
$$
P_D = \frac{V_{IN}-5\,VDC}{I_{OUT}}
$$
where \$V_{IN} \approx 12\,VDC -0.6\,VDC = 11.4\,VDC\$, and \$I_{OUT}\$ is as calculated above (including the >=30% headroom). For example, if \$I_{OUT}=200\,mA\$, then \$P_D \approx 1.28\,Watts\$. If you perform the heat calculations for these voltage and current conditions, and for an ambient air temperature of 28 °C, the 7805's temperature can reach ~116 °C or more (without a heat sink), which is above the boiling point of water!, which could cause the 7805 to overheat and shut itself down. So yes, you're going to want a heat sink on the 7805. (n.b. My "quickie" calculations show you might try a natural convection heat sink (not a forced air heat sink) whose thermal resistance is no greater than 19.1 °C/W. Don't forget to use thermal grease and an electrical insulator between the 7805 and its heat sink, which adds ~1 °C/W of thermal resistance in the heat flow calculations. And be sure to use a heat sink mounting kit that electrically isolates any metal fasteners—e.g., the metal screw—from the 7805's metal tab which has voltage on it! You don't want the metal screw to transfer the voltage from the 7805's metal tab to the metal heat sink!)
