SPOILER AHEAD: A couple of questions only for those who have seen the film

Rating: 10/10 (18 October 2019)

So, I won't comment on the film itself, or Phoenix's stunning performance.I already rated a 10/10 which I rarely do.
The question is: Is really Joker Batman's half brother after all?
At first I thought, no, it is clearly stated that he was adopted. Then my girlfriend reminded me that his mother had said that Thomas Wayne had given her: "some papers to sign".
What were those papers? Were they the adoption papers? Why were they handed to her by Wayne? Why would he bother about a half crazy employee of his?
Most Importantly: How the hell did a single, mentally disturbed woman come to adopt a child? This can only be done if a person of influence, namely Thomas Wayne, pulls the right strings.
The second question is: Did he kill his fictional girlfriend?
It seems to me that although it's not shown directly, it is implied when he leaves her appartment and walks the corridor.
This would make sense if we link this killing with that of his mothers. They are the two people with whom he still has a bond (a real one regarding the mother, a fictional one, yet very strong, regarding the "girlfriend" ). That bond -loving, or being in love with- somebody else, is the one that still keeps the last remnants of humanity in him. Once this bond is cut, the minute he completes the two murders, he is not a human being anymore, even in his own conciousness. It is the time that the transformation into a monster is complete. He has become "The Joker".
So what do you think?