What would my answer be to this question “As a Chabad rabbi, would you have performed a traditional funeral and delivered a eulogy for Robin Williams if he were Jewish?” 
I would say yes.
Why? Because even as Jewish law expresses its categorical contempt for suicide as a viable option it makes clear its empathy for those acting under the kinds of compulsion brought on by depression and other forms of mental illness.The only one for whom suicide is to be regarded as a grave sin is “someone with full knowledge of his actions.” That, rabbinic authorities have agreed, is a standard from which almost all suicides are to be judged as falling short.Rabbi Yechiel Epstein, in his classic work the Arukh HaShulchan (Yoreh De’ah 345:5) states, “This is the general principle in connection with suicide: we find any excuse we can and say he acted thus because he was in terror or great pain, or his mind was unbalanced, or he imagined it was right to do what he did because he feared that if he lived he would commit a crime…It is extremely unlikely that a person would commit such an act of folly unless his mind were disturbed.”
Jewish law has found a beautiful balance between abhorrence of an act before it is committed and compassion for the victims of self-destruction in the aftermath of its tragedy.