MacOS – App cannot create folder when needed, needs permissions

This is actually the correct mode of operation for your OS. This improves the security of the operating system by ensuring that all expanded permissions must be intentionally set, rather than simply inherited. However, you can change this by using Access Control Lists (ACLs) in OS X.

Using the Terminal app in Application > Utilities, navigate to the parent folder in question using the cd command. Then, run the following commands:

chmod -R 774
chmod +a "group file_inherit,directory_inherit" .

The first commands allow everyone in the folder's user and group to access all files and folders in the directory, and allows people outside the group to read the files. The second sets those permissions to inherit to all new files and folders within the shared directory.

For more information on chmod syntax and man page, look here: http://ss64.com/osx/chmod.html

And here is a general chmod tutorial: http://catcode.com/teachmod/

Reinstalling Lion won't necessarily save your data. It sounds very likely your hard drive was failing (disk access substantially slower than normal can be a symptom of failed reads and/or the drive attempting to recover), and it died during the defrag.

The USB drive is only used to install Lion (just like a DVD), not to transfer any of your existing data.

Your next step should be seeing if you can mount the drive on someone else's computer, either via Target Disk Mode, or taking the drive out and putting it in an external enclosure. However if the recovery mode can't mouth the drive, chances are another computer can't either.

It's hard to say for sure without more information, but at this point it's likely your best bet for recovering your data is having good backups. Failing that, there are data recovery services, but they can be extremely expensive.