Do professors discuss about grad students (not necessarily their students)? Do professors discuss about their PhD students or other PhD students and their performance or likelihood of being successful? Grad student community is much smaller than undergraduate community and often a professor has just handful of PhD students under them. So, it is possible the other professors might know about an underperforming or overachieving student from a different group.

If yes, what is the nature of discussions that they have? How often do they feel the need to advise or provide assistance to a student that might have caught their eye?
Yes, faculty do this all the time. They talk about their progress, their attitude towards research, and actively seek advice on how to deal with some situations. This is a positive thing in my opinion, paraphrasing: it takes a village to raise a grad student!