Under normal circumstances, cigarette smoke will not set off a smoke detector.
Frasier's smoke alarm wakes everyone up at 3 am, yet all the buildings outside his balcony are lit up much as they would be in the early-to-mid evening.
The cigarette box Eddie brings in from Daphne's room appears to be a Marlboro red label box. Most women who smoke Marlboros prefer Marlboro lights box.