If I have an opponent's Pokemon Tool F card attached to my Pokemon-EX, does its effect stop working if I evolve into Togekiss with the "Bright Veil" ability and make it my Active Pokemon?
Bright Veil only affects the play of an Item. That means the act of playing it. If it is already in play, enabling Bright Veil has no impact on it. It will stay in play and keep doing what it is doing.
If my opponent has Togekiss with the "Bright Veil" ability Active, does that prevent me from attaching Pokemon Tool F cards like Jamming Net or Head Ringer to their Pokemon?
No, Bright Veil does not prevent you from attaching the Pokemon Tool F card to your opponent.