True love and fairies

It would not unless cognisant and willing. Even then the cords which bind a fairy erodes what friendship, if any.

I think it would involve a long and dramatic story to have a Faerie have true love. Love involves choice, which necessitates will.

Binding a faerie familiar using the Outer Mystery common to the entire house is fine, though the familiar cords degrade over time. The Merinita have found additional familiar cords to bind for faeries, that make it more permanent but cause the faerie distress.

Yes. Faeries can be bound as familiars by those with Faerie Magic; the bond follows standard rules unless otherwise noted, and there is no noted exception to True Friendship arising between magus and faerie familiar.

Your familiar must be willing to be bound with the "standard" familiar enchantment.

As for cognizance, there's absolutely no need for it. I think a lot of people tend to misinterpret cognizance. An uncognizant faerie is not (necessarily) stupid, or without will or self-awareness. It is simply not aware of its "metanature": it is like an actor that does not realize he's acting rather than "living for real". An uncognizant faerie prince utterly believes he is a prince, and does not realize he could just as well be a mad ghost, a clever donkey, or a dragon. So in some ways, it's easier for an uncognizant faerie to have True Love.