Is it ever appropriate for company executives to share their thoughts in various online forums under an alias?
I would argue that it is better for company executives to share their thoughts (assuming that these are thoughts about their own company) under aliases. This will allow much more for a free give and take of ideas. It will also avoid committing the executives to any ideas they might float in such forums.
If executives share thoughts under their own names, they will tend to stifle discussion. People will not want to speak honestly once they have seen what "the boss" thinks. To get better and more honest feedback from others in the discussion, executives should use aliases.
By using an alias, executives also avoid committing themselves to any given idea. Under aliases, executives can propose ideas as "trial balloons" and can see how they are received. Executives will not be committed to actually implementing the idea nor will they be humiliated if the idea is rejected.
The one thing executives must not do is to punish others who give their honest opinions, unaware that their boss is part of the discussion.