Just finished the game. Not sure I understood why Ludwig had a picture of Von Glower in his locket after he killed himself in the lake. Also, why DID Ludwig kill himself and the doctor?

During your visit to the Ludwig Museum you probably noticed all the references to "Louis" and the speculation that Ludwig used that name multiple times to refer to different men who he was close to / was in relationship with. It's clear that von Glower was one of these individuals, and that this was another example of von Glower's unsuccessful attempt to create himself a companion.

Ludwig was bitten by von Glower and becomes a beta werewolf, but frustratingly to von Glower he resists the urge to hunt and kill. Instead he resists the change where possible (which his servants and friends interpret as madness) and after realizing that music has the ability to affect the change, he works with Wagner to construct a trap whereby his music will resonate with crystals to force the change upon von Glower during a public opera.

Ludwig is aware of werewolf law; possibly even from the same book Grace finds in the Schattenjager library in chapter 2. We assume Ludwig studied it when he realised what had been done to him by von Glower. He knows that there's only one way to save his immortal soul: for the Alpha werewolf to die by a Beta's hand. Once he is arrested and knows that his plan with Wagner has failed, Ludwig knows that the only way to end his own suffering is to commit suicide. For a werewolf to be completely destroyed, death must come by one of the four elements. He decides upon drowning as the method he will use.

The death of Doctor Gudden is more of an accident - Ludwig is so intent on ending his own life that he overpowers the doctor to achieve his goal and we assume the doctor dies trying to stop Ludwig from committing suicide.

The locket shows that, although on one level he hates von Glower for what he did to him, on some level he still loves him.