Hi! Little follow up here, now my device is finaly working !
I left it alone for a while, and i started again by segregating functions on separated Arduinos. Now my Arduino Uno only receive bytes via Serial port, telling it to light up some light bulbs (or launch a sequence). Much easier for debug. I made a cleaner installation to avoid power issues.
I also changed ALL the chips because due to my early issues, some had overheated and i didn't want to lose time because of one faulty chip.
The capacitors was a good advice - which is obvious because it is written in the datasheet.
The buffer chips, however, messed with the datas. Maybe i did not use the right ones, but it started working when i removed them. In my case they were not necessary.
Obviously i made a lot of changes, so i cannot say for sure what was my issue. There was probably many, and i might have created some more while trying to fix them. Thanks to all of you for you help and support 
Attached is a picture of my shitty work, and a video of the final result. The sound you hear is an old keypad.
