When I try to add a new file to a zipped archive, I get the message "Operation not supported." Anyone know how to fix this?
There might be a problem with the permissions of the file.
Right-click the file, choose Properties, then Security, and verify that you have the permissions to modify this file. You might also verify you have the necessary permission on the directory, since 7Zip probably needs to create a new archive for this operation.
Summary: Don't press the "Add" button. Use "Add to archive..." context menu from Windows Explorer, or open the archive in 7-Zip and then drag files in from Windows Explorer.
This problem can be you expect the 7-Zip window to work like similar-looking tools (e.g. WinRAR, WinZip) and so expect to be able to press the "Add" button and have it prompt you to select files.
In fact (I think) it wants you to select files and then press "Add". The 7-Zip window is a Windows Explorer/File Manager kind of window, and the "Add" button is the equivalent of right-clicking on a file in Windows Explorer and selecting the 7-Zip context menu item "Add to archive...". So the error is it's way of saying "You haven't selected any file to add."
If you turn on "Two panels" view it may be easier to see how it works, because there you can have the contents of an archive in one panel and the directory tree in the other.
(This is my interpretation based on http://sourceforge.net/p/sevenzip/discussion/45797/thread/0b8b2c38 and trying it out. It seems like that's how it works.)
7-Zip does not support write operations on all formats, there are plenty of formats for which it provides 'Unpacking only'.
Packing / unpacking: 7z, XZ, BZIP2, GZIP, TAR, ZIP and WIM Unpacking only: AR, ARJ, CAB, CHM, CPIO, CramFS, DMG, EXT, FAT, GPT, HFS, IHEX, ISO, LZH, LZMA, MBR, MSI, NSIS, NTFS, QCOW2, RAR, RPM, SquashFS, UDF, UEFI, VDI, VHD, VMDK, WIM, XAR and Z.