When I post, on the left hand side where it shows the summary of my profile, why do I have an icon and text that says "Watched"?  What does this mean?

It means you received a warning for one or more of your posts. It goes away automatically if no more warnings are incurred.

After what time period or after how many posts?  Can I see which post was the offending one so that I know what not to say to prevent this in the future?

violation of rules 2 and 20 of the TOS.  I am not sure what specific threads.  It goes away so many points per day.