The Old One and the Will's description of how to use the Improbable Stair is the same because they're parts of the same being.
In the earlier books, Lady Primus is always talking about how Arthur or herself must use the keys to patch up the holes that Nothing is seeping through in various parts of the house. It apparently didn't work, despite her splitting herself in two, but it all makes sense by the last book because she actually wants the Nothing to destroy everything.
Some of the sins that go with each day match the names very well.
Thursday's sin is Wrath. "Thursday" is a corruption of "Thor's day," who is shown to have anger issues in the myths.
Friday's sin is Lust. "Friday" is "Frige's day." While not much information on Frige survived Christian purges, she is generally agreed to be a sex goddess.
Sunday's sin is Pride. "Sunday" is obviously "Sun day," which brings to mind Icarus, and it is also called "The Lord's day."
The others don't fit as neatly, but might have inspired the author. For example, Monday is "Moon's day," bringing to mind night and sleep, which implies sloth. Saturday fits for Saturn - the Roman equivalent of Kronos, whose envy of the son prophesied to replace him led him to devour his children. It's Tuesday and Wednesday that are the big misfits - Tuesday is Tyr, the one-handed god of war, law, and heroism who doesn't seem to have any connection to greed, and greed or lust (for knowledge or power) would fit better for Odin/Wotan/Wednesday.
Why didn't the Architect just pick one of her own kids to be the Rightful Heir? Sure, the Mariner wouldn't want it, but Lord Sunday and the Piper would probably be happy to destroy the Universe if they knew they'd be allowed to rebuild it. Well, look at the behavior of Lord Sunday and the Piper when pressed into situations where they're required to improvise. When Sunday imprisons Arthur, he ends using a nearly identical version of the Old One's prison. And while the Piper claims to be as creative as any mortal, the Newniths he creates are really just humans with a few aesthetic differences. Neither of them are actually capable of originality.