From Darjeeling (Bagdogra airport), you can take a flight to Kolkata or Delhi. Both have non-stop flights to Dubai via Emirates.
For a 3 day stay, the suggested plan would be -
Day 1 - Arrival at Bagdogra, 3 hour car journey to DArjeeling. Spend the evening at the mall people watching and walking around Observatory hill. Have dinner at Glenary's.
Day 2- Breakfast at Keventers. Darjeeling local sightseeing - HMI, Tea Gardens, Botanical Garden. Have lunch of momo's at Penang / Freedom restaurant.
Day -3 Tiger hill for sunrise, Ghoom Monastery, Toy train Joyride in the evening.
Darjeeling is an unique place. You just have to venture off the tourist beat and explore on your own. Its a very safe place, so don't worry. However, its always wise not to venture into unknown areas at night, you might get lost above anything else.