Jack Lord was upset by Peter Lawford's refusal to pay his 'extravagant' room service bill. He proceeded to claim to have him blackballed from working on CBS shows. The last CBS one Peter did as in 1981 on "The Jeffersons" as 'museum guide (Voice)' in "The House That George Built" (not on-screen though).
Millicent Shand is a mystery author much like Agatha Christie, and her signature character, Henri Renault, is like Christie's famous detective Hercule Poirot.
Peter Lawford was inebriated on the set and kept muffing his lines. In the opening scenes with Mildred Natwick's character, it appears he might stumble into her as they are walking along.
At 46:30 in the freezing room the machinery makes the sound of Star Trek TOS (NCC1701) photon torpedoes.