Andreas,
What is the radio button? You mean the little black point which can be checked? How does it work in general (something I missed out)
The radio button is the little black button as you say, it is used if you just place a symbol and give it only a Part Name property, this is partly legacy from a different product, but it means you can use the same Part Name for multiple Part Numbers and define the default Part Number in this case.
Ah, you mean the symbol need to have all the three properties? At the moment there is only Part name and Ref Des.
No, it is not necessary for the symbol itself to have the properties embedded within it, though it helps to have them in the symbol definition without any values (placeholders) so you can control there position and visibility. But they get added either during packaging or when you place them using the Part View from DataBook. But if you only have one or two on the symbol in the schematic packager is most likely to fail.
It is difficult to determine why packager is failing in this instance because on the one hand you state you typed the Part Number into the property window, it might be that you have an extra hidden character in there, but then you say you placed from the Part View in the library. Two things to try, first, in the library go to the Part Editor and the Part Mapping dialog and see if the Symbol/Cell Preview works. Also post a view of this window on here.
Second, try removing Part Number and Part Label and see if the part packages with just the Part Name property. Part Number and Part Label will get added by packager.
Also show a screenshot of the Property Definition Editor sorted by name and focused on the Part Name/Part Number properties.