Arduino: pins still active after entering sleep mode
The problem is that you are not specifically doing anything to "turn off" the LCD (that is, switch off the back light - note that the LCD module will still be consuming power while you sleep).

When you sleep the MCU it does just that - it goes to sleep. Its clock stops ticking. Not a lot more than that really.  So if you are using PWM to control the backlight of the LCD, when the clock stops ticking the PWM will stop happening, and the I/O pin will remain stuck in whatever state it happened to be in at that moment - leaving you with random on or off at sleep time.

The solution? Manually turn off the backlight just before you sleep.

digitalWrite(9, LOW);


To save more power you might want to consider adding a high-side switch to the power of the LCD module. With that you can physically disconnect the power to the LCD to stop it consuming any power at all during sleep mode. Of course, after re-enabling it you will need to fully reinitialize the module.

The same goes for any other external hardware that takes a power supply.

A high side switch can be built from a P-channel MOSFET being controlled by an N-channel MOSFET. There's plenty of examples on the web.