Game is running fine for me so far, just had to fix the resolution and make the camera not rotate constantly.
To set resolution and window copy ~/Games/Steam/Linux/steamapps/common/Psychonauts/DisplaySettings.ini to ~/.local/share/Psychonauts/DisplaySettings.ini and edit.
To fix the right stick on a controller making the camera rotate constantly add ~/.local/share/Psychonauts/Profiles/Profile [n]/Profile [n]- [nickname].ini and change:
LookUp_Alt=JoyRotZ LookDown_Alt=JoyRotZ LookLeft_Alt=JoyZ LookRight_Alt=JoyZ
to
LookUp_Alt=JoyRotY LookDown_Alt=JoyRotY LookLeft_Alt=JoyRotX LookRight_Alt=JoyRotX
I haven't really found a good button mapping for the 360 controller, as it's not seeing the analogue triggers. At the moment I'm using:
MoveForward_Alt=Up MoveBackward_Alt=Down MoveLeft_Alt=Left MoveRight_Alt=Right Jump_Alt=Joy1 Attack_Alt=Joy3 Use_Alt=Joy4 Cancel_Alt=Joy2 LockOn/Float_Alt=Joy5 PsiPower1_Alt=Joy6 PsiPower2_Alt=Joy8 PsiPower3_Alt=Joy9 InventoryMenu_Alt=DPadLeft PsiPowersMenu_Alt=DPadRight FirstPerson_Alt=Joy11 Journal_Alt=Joy7 Stats_Alt=DPadDown LookUp_Alt=JoyRotY LookDown_Alt=JoyRotY LookLeft_Alt=JoyRotX LookRight_Alt=JoyRotX