It's not in the official wemos store so it's safe to assume that it's a knock-off.
That doesn't mean it's bad, but you won't quite know what you're getting.
the output of the power supply (V+) goes to 5V and the V- goes to Gnd on the modules. The power does not go through the capacitors, as shown in your diagram. The + leg of the capacitor also connects to the 5V connection on the modules and and the -leg goes to Gnd connection on the modules.
you have the data signal from the ESP8266 going to the Din pins on both LED strips. This will work but both strips will display exactly the same LED pattern. If you want them to be controlled as one long string then the Dout at the end of string 1, should be connected to the Din of string 2.
The traces for the LED power on the strips are very small, and therefore have relatively high losses over the 5 meter length. I've read other people recommending that you split the power so it goes directly into two, 2.5 meter lengths. Otherwise the light intensity and colors can vary from one end of the string of lights, to the other.
How do YOU hold it still while you solder the 0ohm antenna resister?
I orient the board so the pinky on my left hand holds the board down, while I manipulate the shunt with tweezers held between my thumb and forefinger. I hold the soldering iron with my right hand.
It is tricky. While I was doing this on one board the resistor went flying, never to be seen again. I used a small piece of wire to bridge the pads.
It works when connected to the PC's USB port, but not when running from wall power. The element that changed is the power source, so I'd start with verifying or fixing the power source; especially given the high ripple voltages found on many power sources.
The first thing to check is the quality of the power coming from the charger/power supply. Cheap power supplies often produce lots of noise. If you can, check the power input to the WeMos module then do so. If you don't have access to a scope, try putting a large (e.g. 680uf - 1000uf) capacitor across the power and ground pins.
SPI is faster than i2c so it's great for things like graphical LCDs, where you're transmitting 18-24 bits of color data for every pixel. But the amount of data moving out of a BME 280 is so small that I prefer i2c because it uses fewer I/O pins than SPI.
FYI, I haven't done any electronics or ESP8266 tinkering since the spring but when the weather gets cold, I'll probably get back into it.
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