B. Let's wait 'til Mom gets home.
If you chose A, count yourself among the countless millions who have forgotten that the third word is actually not a whole word. The confusion stems from the relaxed way we speak. "Until," when said quickly, usually comes out sounding like "'ntil," or "til," hence the error.
B. Let's eat, Grandma.
You've seen this before, usually under the title, "Punctuation kills."
The first sentence is something a hungry cannibal would say. The second one lets Grandma know the speaker is hungry.
A. Safe deposit box.
"A" is the correct choice here. Again, lazy speech patterns have led to this extremely common mistake. The box in question is one that is located in a safe.
B. The P90X exercise regimen is the best I have ever done.
A "regime" is a form or mode of management, à la the "Hitler regime."
A "regimen" is a way of life, a prescribed course of medication, a practice. Even the best of writers get this one wrong, and it's sad, really. Writers should know better.
By the way, "regime" is pronounced re-JEME, not RAY-jeme -- unless, of course, you are French.