A brilliant energy weapon has its significant portion transformed into light, although this does not modify the item’s weight.
A brilliant energy weapon is not purely made of energy or anything like that, it simply turns the part of the weapon that deals damage into energy (to be honest, the fluff is up to your GM). So, for a sword, that would turn the blade into energy, while the handle is physical. For an axe, the haft would still be physical and maybe it´s connection to the blade. For an arrow, this would most likely mean that the tip is made of energy. And so on.
Without such a connection, you couldnt possibly drop the weapon without it going through the floor.
A brilliant energy weapon ignores nonliving matter. Armor and shield bonuses to AC (including any enhancement bonuses to that armor) do not count against it because the weapon passes through armor. (Dexterity, deflection, dodge, natural armor, and other such bonuses still apply.) A brilliant energy weapon cannot harm undead, constructs, or objects.
The bolded part is what has merit here, since cut from the air clearly says that you use your weapon to cause this. But this is simply fluff text. Nothing really prohibits the character from using cut from the air with his bare hands, if he is able to make attacks of opportunity with them. And if he does use his barehands (or barefeet for kickers), he completely negates that text from brilliant energy weapons.
Also, the text goes on to clarify what exactly means to ignore nonliving matter, explaining that regular armor and shields offer no protection against brilliant energy weapons. But that's it, trying to stretch this too far is asking to break the rules and walk straight into house ruling.
And to prove that, we can see that it doesnt ignore AC bonus from cover, meaning that regular cover, be it a low wall or another character in front of you, still offers an AC bonus, even if this bonus comes from a source made of nonliving matter.
When this translucent +2 arrow is fired, it becomes ethereal until it enters the target’s space. A phase arrow ignores cover of all kinds, including total cover, and can be fired at a target even if the wielder doesn’t have line of effect.
Well, this one is simple, since this has been asked on this site before. We know for a fact that a creature can threaten and attack on their own space.
You can attack into your own square if you need to, so you can attack such creatures normally.
Since the arrow is no longer ethereal, it can be deflected just fine.
Does Smash From the Air work against Brilliant Energy or Phase arrows?
Yes, but cut from the air is actually enough against both types of arrows, as none are transformed into spell effects. They are still real weapons/ammunition that are transformed and enhanced by magic.