I have recently installed Cygwin along with the Git binaries and a copy of TortoiseGit on a fresh Windows XP install. I seem to be able to do most Git operations through Tor
The "unable to create temporary sha1 filename" issue was solved for me by:
https://stackoverflow.com/a/685422/431606
I simply ran git repack
(from the command line) in my repository, and TortoiseGit was immediately able to commit.
(I still can't push through TortoiseGit, as I get the error message "fatal: The remote end hung up unexpectedly" as described in https://tortoisegit.org/issue/56)