Honestly your choice of hardware is your main issue. If you had used an ESP8266 based board and individually addressable LEDs like i mentioned before your code would be only a few lines.
Like for the LEDs you include the fastled library, inintialize the LED array replacing NUM_LEDS with the number of LEDs you have connected.
Then you setup it up with a single line of code. Replacing LED_TYPE with your LED model number, DATA_PIN with the pin you are using for the LEDs, COLOR_ORDER is replaced with GRB or RGB etc. The last thing is NUM_LEDS is replaced with the total number of LEDs being used.
FastLED.addLeds<LED_TYPE, DATA_PIN, COLOR_ORDER>(leds, NUM_LEDS);
Then to write to an LED you simply type:
leds[LED_NUMBER] = HUE;
You replace LED_NUMBER with the led you want to change the color of and replace the HUE with the hue you want from 0-255. You can also address the colors as named colors like blue, red, aqua with a slight change. There are lots of examples on how to use the library. You can set all the LED colors before calling fastled.show and it writes out all the LED data in a single go. There are also functions to set all the LEDs the same color in a single go or to fill them with a rainbow of colors etc.
Then for the servos because the ESP boards can use almost all the pins a PWM pins you can just simply write a PWM value to them like you are doing with your first 8 servos on the Arduino.
If you insist on using your current hard to use hardware may I suggest you at least look for libraries to make coding easier instead of trying to write it all yourself. Like for the I2C stuff you can use the wire library from Arduino or many other serial libraries available online. I am sure you will also find libraries for your shift registers and servo board etc.
EDIT: I also have boxes of hundreds of discrete LEDs and would never consider using them for anything other than indicators due to the time it would take to connect and code them vs the time it takes to connect 3 wires and write simple code for individually addressable LEDs. I would say this is even more true for someone new coding and/or hardware