So Ysgrammor ruled from his throne in Windhelm. His line continued on down to King Borgas, who was killed by the Bosmer's Wild Hunt.

1. So Ysgrammor's bloodline is lost forever, or do some of the Jarls still have traces of it? I assume Borgas was a direct descendent (like a son, of a son, of a son, of a son of Ysgrammor), and it's possible that that someone else had that same bloodline.

2. Do the Companions technically have rights to Skyrim since they were founded by Ysgrammor?

3. Why was the Wild Hunt started at all? Why were they in Skyrim?

4. Who ruled after King Borgas and how did they come into power?

1. I'd say it is extremely likely someone may still have some traces of Ysgramor's blood in his/her veins, however there wasn't anyone officially recognized as a (distant) heir of Ysgramor since Borgas.

2. Not really, seeing how they didn't get any rights from him - if the Queen of England were to found a foundation, it wouldn't get the rights to the kingdom after her death, would it?

3. Borgas wanted to join forces with Cyrodiil and attack Valenwood, his death didn't take place in Skyrim - he got killed during his mission in the central province.

4. Olaf One-Eye, but before he got the throne, the War of Succession was fought for 50 years. How did he seize it we don't know - it is likely however that it had something to do with his capturing of Numinex.

Didn't you help a woman find a scroll in some generic draugr dungeon which proved that her family were descendants of Ysgramor?

EDIT: http://www.uesp.net/wiki/Skyrim:A_Scroll_For_Anska

Of course, direct blood descent from Ysgramor is nothing entirely rare nor does it give a person any political authority. The line of legitimate ruling kings descended from him died out long ago. The remaining blood relatives are all from splinter families. It's like the real world examples of being descended from Genghis Khan or Charlemagne. There's millions who can claim that, but it doesn't mean so much any more. That ship has set sail already.