How to design blinking LEDs circuit using analog elements only?
555 is not a right component for the task; it is okay to use for two LEDs, but things quickly get hairy for more than two LEDs. You need a Ring Counter + astable multivibrator (Examples of ring counters can be seen here: https://www.google.com/search?q=ring+counter+circuit&amp;source=lnms&amp;tbm=isch). 

You can try this one for example: http://www.tradeofic.com/uploadfile/ic-circuit/20097142649472.gif

Keep in mind, most of the schematics are ancient and use antique components, so you'll have to find modern equivalents.