How the breakdown voltage of zener diode is set during manufacturing

It's doping. See Breakdown voltage - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Breakdown_voltage).

Actually the mechanisms are Avalanch and Zener. Both mechanism are occurring at 5.1 V. The diodes are all called Zener even though some are technically not Zener diodes.

Breakdown of diodes is typically non-destructive.