When I try to add a loop cut by pressing CtrlR only a dot shows on my object instead of whole loop cut line. What am I doing wrong to cause this problem?
Try selecting that lower edge and moving it down a little. If it is not connected then just delete it. If the edge you select is connected but you see that there was another edge hiding below it then select that edge and move/delete it.
Or you could select everything and from the Tools panel on the left and use the Remove Doubles tool.