Eureka! I've managed to make my first android project import to Eclipse!
Earlier I always got "Unable to find system library" under the project's Properties - Java build path - Libraries tab and it was impossible to correct it once it occured. The consequence was that no import statements could be resolved by anything imported.
In this case, I changed the JDK compliance level from 1.6 to 1.5 BEFORE I imported (afterwards its too late and cannot be mended). It must be done from Window - Preferences - Java - Compiler (earlier I thought this option was available only by right-clicking on already imported project). Then right-click on each project and set Properties - Android to a suitable Target (with the errors above, eclipse ignores any selection here). Then, I could even change back to JDK compliance level 1.6.
So, in general, changing the JDK compliance level before importing, might help.
And when deleting a project ALWAYS check the "Delete project content on disk (cannot be undone)" checkbox in the "Are you sure" dialog. Otherwise it is impossible to re-import the project, o if you've imported it with the wrong JDK compliance level the first time, it is impossible to correct the mistake, other than to try in another workspace.
These are the experiences of a very frustrated Eclipse user.