Re: Which female character shared a deeper connection w/ Bruce?
Rachel, hands down.
Selina showed up in the last act of the trilogy, and so didn't have much time to get to know Bruce, whereas Rachel had two whole movies with him.
Selina and Bruce had little screentime together, while Rachel and Bruce knew each other since childhood.
Rachel helped show Bruce the difference between justice and revenge, and was kind of a moral compass for Bruce.
Her letter in TDK foreshadowed the finale of TDK, when The Joker had the two ferries ready to blow. Batman didn't believe the people would blow each other up, because he had too much faith in them, as her letter said: "Don't lose your faith in people." Bruce didn't read the letter, but still, there's a connection there.
But then again, who cares if Bruce spent his whole childhood with Rachel, who cares if Rachel was his only true friend? Who cares that she is the only person that ever really understood him? Who cares that he pined for her all of his life, and learned many lessons from her which helped mold him into Batman? Selina is a sexy babe in a tight outfit! She was in the comic book, and Rachel wasn't!