Its really hard to say when shops close. There are malls with grocery stores that close around 11pm or midnight. They open at 7 or 8am. Most shops close on Fridays or have limited hours for Prayer. There are several villas around the airport area on between D-Ring & C-Ring. Be careful with the sound from overhead planes when living close to the airport. West Bay is not far at all from the airport. You'll be going against traffic during rush hours.

Good luck, and I hope you find the answers to your other questions.