Is the LCD I2C or serial (does it connect to A4 & A5 or does it connect to RX & TX)? How about a link to the LCD datasheet? Which I2C lcd library, there is more than one? Define trouble, like no display, row of black squares, etc.
The external interrupt flag register, EIFR, has a flag for each external interrupt. You should be able to check that register to see which pin caused the interrupt. Check the data sheet under external interrupts.
The Nano can easily take care of blinking the turn signals. No need for relays anywhere as you can drive the lights with MOSFETs controlled by Nano digital outputs. Pick your LEDs. Once you know how many LEDs and their current requirements you can specify the transistors and power supply(s).