before attempting to answer your questions, I really have to know your familiarity with Photoshop. Have you ever used the application? Have you read the documentation, the Help files or any books?
Maximizing Compatability includes a composite file in addition to your Layers; and makes it possible for older versions of Photoshop and other Applications to open your files.
It does make the files a little bigger but I choose "Always".
I save Icons and Thumbnails (but not the full-size Preview) and append lowercase extensions.
You definitely want to append the file extension ALWAYS.
I suggest that you leave them checked and with "Always Save" —except for the Full Size preview.
Check out the Photoshop Help features (such as this one) for more info.
<http://help.adobe.com/en_US/Photoshop/11.0/WSfd1234e1c4b69f30ea53e41001031ab64-7720a.html>
I would just leave them switched on so that they are there when and if they are needed — either by you or by others.
I am not certain but I think that e-mail attachments may use them for example, and the Mac Finder does too.