1. When does the timer start?
The timer starts when the offer is received, not when you start collecting items for that quest.
2. The timer showed that I still had 20 more hours, why did the quest disappear?
Please note that some quests have 2 different kinds of timers: one for the overall quest and individual timers for each stage of the quest. Please do not confuse the 2 kinds of timers: the one for the overall quest is located on the right side of the screen, the individual timer for each stage is displayed in the upper left corner of the stage window.