I'm having the same problem. Mac OS X 10.6.8
Delete key does nothing, using both Direct Selection and Selection tools.
Hey I figured it out!
The problem is that when I upgraded, I kept my previous keyboard shortcuts file. Most of that worked OK, but some shortcuts got corrupted.
To fix, I had to go in and reset all features that were behaving improperly.
So for the simple delete, you have to actually assign the Edit > Clear command to the delete key again.
Same for a soft return, which I think was under Type > Insert: Forced Line Break (this one has an added drop down menu from which you have to choose "Text" as well).
Bless you, that works. Much appreciation.