Well, it was opening in read only mode and in read mode (meaning with the arrows). Then, I Googled a bit, and Microsoft said to:
Go to File | Options | General and uncheck the Open email attachments and other uneditable files in reading view. So, it does now open in Edit Document "view" but I still have to click on Edit Document in order to edit it. (even hit the return key).
I read somewhere to uncheck the three boxes under Protected View in the Trust Center. Did nothing for this.
I should say that my Word documents on the client computers open in Edit Mode. It is when I open it from SharePoint that I have the problem.
FYI: I am leaving work to go home, so I may not see your answers until tomorrow, but I will get back to you promptly.