Diode in Parallel – Purpose of Diode in Parallel to the Fan

I think you've got it mixed up. The body diode is a part of the structure of the MOSFET and does not help when switching an inductive load with a single MOSFET.

The zener in the symbol represents the avalanche rating of the MOSFET and is of use when switching an unclamped inductive load. You won't see it conduct unless you exceed the voltage rating \$V_{DS}\$ and get to \$V_{(BR)DSS}\$.

If you are actually getting +/-12V out of the pulse transformer, I don't see any obvious reason why that part would not work. Your negative swing out of the pulse transformer does have to be sufficiently large to turn the control MOSFET on, when loaded with both gate charges (-10V would be good).

I understand that reverse diodes inside MOSFET package are put there to protect the transistor from inductive reverse current.

Actually the internal diodes in MOSFETs are a not there explicitly for that purpose. Normally MOSFETs would be a 4-pin device with source, drain, gate and body terminals. But because these are "not that usefull" and the body terminal could cause inconvinience forming a PNP or NPN transistor inside, this gets shorted to the source preventing that. Thus the body diode gets formed.

Why to double reverse diodes in parallel?

The external diodes are used in this schematic, because the used MOSFETs have a high turn on voltage for their body diodes. N-channel has a body diode voltage of 2.5V and the P-channel has a body diode voltage of -6.3V. Meaning the voltage would have to rise above 2.5V or below 6.3V for the internal diodes to start conducting. This could be a problem and damage the devices and I guess the designer just want to be completely sure nothing bad happens.

Maybe external diode can be ignored if the transistor packaging includes the reverse diode? Maybe some special diode should be place in addition to MOSFET internal diode?

Usually the internal diodes are sufficent. But if they have a high turn on voltage such as your P-channel and even your N-channel has relatively large turn on voltage, external diodes are placed like in the schematic. If speed is important in your project, I would recommend using schottky diodes as these are much faster than normal, and have a lower turn on voltage.