Val will probably have a better refutation.
Not really :)
I had always taken the name Morgul to mean just "Dark Sorcery" or "Black Arts". The Witchking was known as the Morgul lord, the Morgul-king and the Lord of Morgul. These names were given to him before the capture of Minas Ithil, but I never thought of the city being called Minas Morgul after him specifically. I always assumed it got the name from the fact it was occupied by the Nazgul in general.
While looking for a better answer for this question I did come across a piece in Tyler's Complete Tolkien Companion which does seem to back up Aule's original claim (although this is possibly where Aule originally read the statement???)
Morgul "Black Wraith"(Sind) - In the absence of clear information from surviving records of the Third Age, all hypotheses must assume the lesser stature of guesses; nonetheless, it seems possible that Morgul was the name or title of the Lord of the Ringwraiths, the Witch-king of Angmar, who later became the Lord of Minas Ithil, which his armies captured on Sauron's behalf in 2002 Third Age. The Tower of the Moon was then renamed Minas Morgul, and the vale in which it lay became known as Imlad Morgul.
Personally I think this claim is a little tenuous at the best and if anything, Morgul was a title rather than a name.
The true name of the Witch king is Er-Murazor
I don't think I can refute this as I think it was something I wrote myself in a post ages ago anyway.
The Witch-king, as far as I know, was never named by Tolkien. He was said to be a Sorceror King of the Black Numenorians, however. Judging from the time the Nazgul first appeared, a strong candidate for who he really was, was the second son of Tar-Ciryatan, thus the younger sibling of Tar-Atanamir. Tar-Ciryatan was the first Numenorian to treat Middle Earth as a place of conquest, and it was during his reign that the Shadow was said to have first fallen on Numenor. Tar-Atanamir was the first Numenorian King to cling onto the sceptre until his death, rather than following the tradition of passing it onto his son before becoming aged.
From the dates and the family background, a younger brother of Tar-Atanamir would seem a strong candidate for Sauron to corrupt with a Ring. ICE's Lords of Middle Earth name this individual as Er-Murazor, although I'm unsure whether this name is one of their own making, or whether they have found it in one of the HOME books.