You definately wouldnt feel a unbilical cord.do you even know what it would feel like.at this stage you would be so early,you wouldnt feel anything,let alone see anything on a scan,what you may have experienced with the 1 day bleed,is not a period,since this dont happen but maybe implanation bleeding.or somthing else.i would wait a few days and take a pregnancy test.
I don't think you will be able to feel the umbilical cord. You are probably just feeling your belly button.
I think it is possible to get a bit of a period and still be pregnant. My friend had 3 full periods before finding out she was 3 months along.