What is this keyboard connector and how to connect it to MCU?
What's the common name of this flexible cable connector, if any?


Flex cable connector, I'd say. 


  How to turn the keyboard into a usable one with an MCU (i.e. how to connect it to Arduino or Raspberry Pi, or any other micro controller/board)?


This connector probably just connects to the dumb switch matrix. Keyboards generally work by having "rows" and "columns", and they just power e.g. one row at a time and then look which column is powered (and then scan through all rows). The pressed button connects a row with a column. To make this even more compact, a lot of keyboards incorporate diodes – that way, it makes a difference whether you bias the row with a positive or negative voltage, and you can have twice as many buttons for the same number of contacts.

What you hence must do is reverse-engineer this matrix. This will take some (significant) time to do manually, but you can also write a program that scans all possible in/output and voltage combinations while you press a key.

Does your keyboard by any chance have 88 keys and no LEDs? Because I count 11 lines on one connector, and 8 on the other. If that's the case, at least you don't have to worry about diodes...

Personally: It's probably not worth the effort. Get a working keyboard that feels just as awesome :)

Unless, that is, you can peruse google to find "keyboard matrix" + device type to find someone has done this before you.

Also, you might be lucky and be able to find the keyboard controller IC of the motherboard. That'd solve that issue – simply use that IC, if you can still buy it, or if you can get a datasheet, use that datasheet to help you reverse-engineer your matrix.