OK
DCIM is there for the stills. It's empty on my card. I've never seen a file in MISC as well. I'm able to import from folders containing only the PRIVATE folder, but only unless this copy was made in Mountain Lion and PRIVATE appears as a package.
I am lucky - the card has not been erased somehow. I was able to get the material in straight from the SD and then started experimenting.
Here are my findings. They might be useful to someone later:
1. Making an EXACT copy of SD does not make it 'importable' in FCP X 10.0.5.
2. Besides DCIM, MISC folders there are MP_ROOT and hidden AVF_INFO folders and they don't matter as well
3. To use the archives in FCP X 10.0.5 on Mac OS X 10.8.1 you have to adjust your workflow and actually CREATE archives using the FCP's Import dialog by pressing the button next to Import Archive.
4. All folders, containing PRIVATE folder in them, that have been created before Mountain Lion DO IMPORT without any issues using the same Import Archive dialog of FCP X even under Mountain Lion
With this I must admit that I've found no working solution to the issue, but at leasted I was able to experiment and clearly identify a working process for the future.
Thanks for your help, Tom and Karsten