Whenever I shut the log window, if something is posted to it, it reopens every time.. If I shut the log window, it means that I don't want to see what it has to say, and it should remain SHUT!!

yeah, similar issue was recently reported about file transfer window.
What can I say ? It will be improved.

I would like the ability to turn on/off the various panes as well. For example, if I don't want the whole right-hand side of the window, which has the DB explorer, help files, etc -- turn the whole thing off.

In this case you'd press [x] button on the right top corner of corresponding windows.

Just for your info, the problem with unwanted activation of log window is fixed in phped 3.3.