L9110 12V Getting hot
"The poor datasheet" states a typical  VHin of 5V , up to 9V when powering the board with 9V. Very nice. At 12V it might be higher leaving the driver in the linear region (half opened). 

I couldn't find a real design that uses 12V for the motor driver and 5V for inputs, most were using 5V for the motor to. 

The pull-ups you removed were pulling the Arduino outputs (when high) to 5.5V limited by the clamp diodes, by removing them the the high output is lower and this might be the explanation why your board is not working.

It's not wise to let the inputs without a pull-up, at some point Arduino outputs might float which can damage the driver. Even connected to 12V the current through the pull-up resistors is low and harmless for the Arduino board.