If you put an Aura on an opponent's permanent, you still control the Aura, and it still counts for your spells that have affinity for Auras.

If a spell has multiple instances of affinity, each one applies. For example, if you somehow control two Pearl-Ear, Imperial Advisors and you control two Auras, each enchantment spell you cast will cost less to cast.

An Aura controlled by another player does not cause a creature you control to be modified.

A creature with a counter on it is considered modified no matter what kind of counter it is or which player put it on that creature.

A creature you control that's the target of an Aura spell isn't modified unless it already has a counter on it, an Equipment attached to it, or an Aura you control attached to it.

A creature that is equipped is considered modified no matter who controls the Equipment that's attached to it.