1) You can choose one of your opponent's unit, as the ability does not state that. However, if the resolution requires you to call a unit from a certain clan (eg. search you deck for a <Royal Paladin> Unit), the skill will still require you to do so.
2) AUTO abilities that last until the end of the battle will resolve immediately as the unit is not in battle. The ability would be resolved as end of that battle.
3) AUTO abilities that state that the boosted unit will gain power until end of turn will result in the unit not gaining power as Bizen is not the unit being boosted.
4) Yes... the skills of Phantom Blaster Overlord, Majesty Lord Blaster, Illuminal Dragon, Dragonic Overlord the End can be copied. However, Illuminal Dragon and The End's skill would activate during main phrase, rendering their skill useless, unless you have units that are at rest for Illuminal Dragon.
5) Skills which requires the unit to be in a certain zone to activate will activate, as all conditions for the skill has been met, including the condition for the unit to be in that certain zone.