L293D Brake Doesn't Work (LB1836M good alternative)?
There is already a lot of good information given at the forum link that you posted. I think it is possible to conclude that the L293D is not a great driver for performing the motor braking operation. The output diodes and Darlington output stages leave the motor coasting when the voltage gets low enough that it cannot bias the output stages. You really need a MOSFET type driver that is specifically designed for motor breaking that can hold the necessary low impedance connections to GND on both sides of the motor all the way down to near zero volts. 

Your question "Does the LB1836M make the motor immediately stop if you send 0V to both input pins?" cannot be answered in the affirmative. Simple inertia with practical physical laws says that there will be no immediate possible. 

The LB1836M driver chip will be better than the L293D when it comes to trying to do a braking. This device is still made with bipolar transistor circuitry so will still not be as effective as a driver setup with MOSFET output transistors. 

Note that there are driver chips from companies such as Allergo and others that have all the bridge control logic that you then connect to your own output MOSFETs to form the necessary H-bridges. I've used a some of those in designs and liked the results. 

Since you are anticipating the use of a microcontroller to drive your device you may want to consider the idea of braking through the use of reversing the drive to the motor through a PWM technique. This was discussed in the thread that you linked and can be a suitable way to quickly slow your motor using even the bridge driver chip that you have now. You would just have to make sure that your microcontroller has the necessary timer logic to generate the PWMs and that you can prepare the necessary software to profile the PWMs being generated into the necessary deceleration profile for your motors.