Its not dual typed though. Look at the type listing for the actual move. Its listed as Fighting only. Its coded to deal Flying type damage as well and that's it. The Damage is 2 types, but the Move itself is only 1 type. Hawlucha doesn't get STAB from both types as far as I'm aware and if it were two types STAB should have increased the damage a lot more than that.
If you change the typing of the attack, it still does Flying damage on the side due to the Flying damage is an added effect and not its actual type.
If Flying Press is used while under the effect of Electrify or Normalize, damage is calculated using Flying type and Electric or Normal type modifiers instead.
While it still does x4 damage to a Pokemon such as Abomasnow, this is because both Ice and Grass are weak to Fighting and Ice. If changed to Electric through the use of Electrify, the damage against Abomasnow would be neutral damage since it does change what can be effected by it, not to mention that it loses STAB bonus since its no longer a Fighting type attack. Same if Normalize effects it.
As for Freeze Dry its similar because both attacks have abilities that change the basic dynamics of the type chart's resistance and effective checks. Freeze Dry changes Water's resistance and deals Super Effective Damage while Flying Press adds in Flying Type Damage to off set the Fighting Type damage it does and creates a special list of damage calculation changes for it. Both are due to its effect and not its initial typing,