1 - There are a few short term options in West Bay. Try Ezdan Suites (not great quality, but central with good facilities...fine for short stay) Retaj Hotel have a selection for short term also. Both have studio's, one and two beds, I think budget around 10,000 for a month.
2 - Pick up a Q-tel sim and Hala card (pay as you go) at the Airport
3 - Bundles are available, only a couple of options and readily available. Q-tel have offers.
4 - No bank really stands out. HSBC and QNB have never caused me issues, though some won't agree. Commercial Bank is ok, as is Mashreq.
Hope that helps!! Good luck!