Can I store settings on a nano through itself?

In other words does this chip/board have a flash or other type of NV ram or EEPROM that one could say, adjust LCD contrast and store that PWM value so it stays through resets.

You can save it in the Nano's EEPROM using a little bit of code. Check eeprom.put and eeprom.get.

There are also read and write methods, but the get and put take handle different variable types while read and write only work with bytes. From that perspective I find them easier to use.

scroll down the Nano shop page a go to Documentation tab