"... And Dawn's Beauty, the princedom of Lorkhan; misnamed Tamriel by deluded mortals."
Why does Mehrunes Dagon claim it?
In the Nordic myths, specifically the http://www.imperial-library.info/obscure_text/aldudagga.shtml. Dagon was a leaper who got banished to Oblivion for stealing parts of the world from Alduin. He won't be able to become a Leaper demon untill he destroys all these stolen parts. Because Lorkhan was making the mortal plane out of these stolen parts, Dagon wishes to destroy the mortal world.
In Mankars own http://www.imperial-library.info/obbooks/mythic_dawn_commentaries.shtml tells that the Aedra who according to him were Lorkhans lieutenants rather then contributors to the mortal plane. Because they wanted to dispose of Lorkhan but the concept of Rebelgion didn't excist yet, they created Mehrunes Dagon the Prince of Revolution (amongst other things) from random parts they could find.
Both are similair in the sense that Mehrunes became what he is by the meddling of the Aedra and in both cases Mehrunes has a claim to Mundus. Either in executing the purpose of his creation or to return to what he once was.
Sort of, but that speech has lore inconsistancies in it so its not a really a relible source. He claims that the Coldharbour of Meridia, Peryite's Quagmire, the ten Moonshadows of Mephala are their realms while its really Coldharbour to Molag Bal, Quagmire to Vaermina, and Moonshadow to Azura. So I doubt what he says are really true.
Those were the mistakes of the devs and they slipped past quality control. They've apologized for that.
Mankar isn't all that wrong really and considering he has Mehrunes Dagon at his backing, he might be telling the truth. After all, everything we know comes from the Aedra and if they have been lying, we would have no way to tell.