Could someone please tell me if it`s possible to share from ArcGIS Desktop to ArcGIS Online a .gdb witch has some feature classes, tables and some relationship classes so that I can check my data in ArcGIS Online (also to be able to see data that is related with relationship classes) ?
It is not possible when you have relationship class. Otherwise, it works fine. If you need to publish feature layers that have more than 1000 features, you need to use feature services.
To add to the last comment, all tables that associated with featureclasses with relationship class should have at least one record (your related tables cannot be empty). Otherwise, the web map will not display features at all.