The only thing that has been proven is that many asian men share a same paternity line (0.5 % of total population in the world).
The study suggest that particular mutation came from Mongolia a thousand year ago.
A paternity line of this scale is uncommon. The probable explanation is natural selection.
So the scientists suggest that it should be Gengis Khan line (a male ancestor from Ghengis but not Ghengis himself).
Because this line was highly popular in most Khanates, they blended well in Russia noble men.
Many muslim dynasties in Asia had blood coming from the Khan. These dysnaties commonly had harem.
For Ivan it cannot be proved he has DNA from Ghengis. Not enough document have survived.
There are speculation that British Queen Elisabeth II may be a descendent.