What components do I need to add to the power circuit for this ESP8266 project?
This is an interesting controller method that I have not seen before. It seems reasonably well thought out and your change to ESP8266 seems OK. 

I question the trigger current selected for your opto-Triac, and think the 330 Ohm will not work. 

If you examine the MOC-30xx datasheet you will notice the last digit selects the sensitivity of the trigger circuit. The MOC-3042 you selected has a maximum trigger current of 10mA. With 330 Ohms series resistor you will only get 5mA at best, so the device may not turn on reliably.

 

You need to use the MOC-3043 to get this level of sensitivity. Even then I'd suggest the series resistor should be 270 Ohms with the MOC-3043. 
Note that the original circuit you linked to also has this problem, they were not guaranteed to be able to switch on the MOC-3041, which could require 15mA to turn on under worst case conditions. 

Your selection of the MeanWell power supply seems appropriate. I'd be tempted to provide a power supply reset controller to ensure reliable start/reset of the ESP8266. Perhaps a simple unit such as the TI TVL803S would be appropriate. 

UPDATE:
The issue of input surge protection is somewhat complex when it comes to very small PS units. Your suggestion of MOV's won't work (they are for limiting surge voltages not limiting initial current surges).
From the MeanWell datasheet you can see the input surge is a maximum of 5A @ 120V and 10A @ 240V, this would seem to indicate they already do have some form of current surge protection built in.    

If you want to use an inline fuse, then I'd suggest you need at least a 5A fast blow with a series current limiter. You could use a simple resistor of around 20-33 Ohm, or something like an Anatherm SL10 50002 surge limiting resistor. Note that if you use a surge limiting resistor it won't get hot enough to drop the resistance by much, and so is acting more like a simple series resistor. I would use the SL10 because it's rated for the voltage and help ensure the fuse will blow on a fault. 

You can explore the available inrush current limiters on Digikey.