Mark Studdock is the Patient from the Screwtape Letters
The Tramp whom Merlin encounters is Radagast
Out of the Silent Planet comes from a different continuity than the other two booksOut of the Silent Planet contains a "this is real, but all the names have been changed" disclaimer at the end. In Perelandra, Ransom's name truly being Ransom is a plot point.
That Hideous Strength was started before the events in it were finishedThus, talk of a certain college in the present tense.
That Hideous Strength was written long after the events of the book.Possibly After the End, depending on whether it was truly too late for Jane to fulfill a certain prophecy.
Jane continued to have clairvoyant visions after the events of the book.And she passed them on to C. S. Lewis. (Literary Agent Hypothesis.) That's how Lewis the narrator can know what various villains were thinking just before they died: it was revealed to Jane in one of her dreams.