With my setup I can open an old 60 minute iMovie HD 6.0.3 project with numerous clips. I successfully did a few straight cuts, updated a title and saved the project.
So it seems to work. I haven't yet tried to export/import tape or to iDVD.
I have installed OS X 10.9 from scratch to a test partition, if that matters. Maybe an old install behaves differently?
You might try to remove all 3rd party plug-ins from iMovie as well as from the system. Or just copy the OS X 10.9 installer to an USB stick, boot from it, and do a fresh install on an external disk (MacWorld has instructions for this).
p.s. just to clarify:
You can't install iLife '06 and iMovie HD 6 from the original DVD because the installer can't run in OS X 10.8/9. So you must:
a) 1st install iLife '06 while booted in Mac OS X 10.4-6. Then copy or compress /Applications/iMovie HD.app and /Library/Application Support/iMovie to a safe place (do the same for any of your 3rd party plug-ins -- do a "Get Info" to see if they are "Universal" i.e. compatible with OS X 10.8). Then boot into OS X 10.8/9, and copy the previously mentioned folders to the same folders. Or:
b) Use Pacifist ($20) to open the iMovie HD 6 install packages from the iLife '06 installer DVD (iMovie.pkg and iMovieThemes_Tiger.pkg) and let Pacifist install them to the right places.
iLife '06's iDVD 6.0 (or v6.0.4) does not work in OS X 10.8 so I had to install iDVD 7.0.4 (and iPhoto 8.1) from the "Applications Install Disc" DVD that came bundled with my Mac mini. Remember to choose "iDVD extra content" which installs iDVD 1-5 themes in addition to the default iDVD 6-7 themes. No need for Pacifist with this install.
iMovie HD 6.0.2 and v6.0.3 as well as iDVD 7.1.2 and iPhoto 8.1.2 updaters also work without Pacifist.
AFAIK iMovie HD 6.0.4 is identical to v6.0.3.