WS2812b interference and bootup behavior
First of all, you have a problematic power circuit here, to say the least.
I don't know where you got the schematic or ideas for it, but not only does it give you about 2 seconds power-on delay, it also drops your input voltage significantly (could be anywhere from min. 2V to as much as 8V), and this is the reason for your slow power-up time. You need to switch it on 20 times because it takes time to build up the minimum working voltage. This slow build-up is what causes the Arduino MCU to misbehave.  

My first suggestion is to get rid of this whole circus and just use 7805 voltage regulators FOR EACH Arduino board SEPARATELY (because a 7805 can withstand significantly higher input voltage than an 1117, and because its output voltage can't be guaranteed to reach its destination intact with a long wire between its output and the loading circuit), with 100uF capacitors at their inputs and 1uF (or 10µF; you could use up to 100µF) at their outputs.  

I am also suspecting the ground or common side power supply connection. You may have a ground loop, some resistance (loose connection) anywhere on the ground side, or both. Voltage reference points, ground loops and signal integrity are all a tricky business, especially if you don't know about them.
Do you know the precise timings of the signal to the LED strip?

Please provide a photo (or photos) of your setup so that we can solve your problem! Without seeing the actual setup, we are sort of shooting in the dark here.