1). Inspections: No inspection contingency is preferable to the bank, but if you have one (and I'd urge having one), make it short, i.e. 5-7 days.
2). Financing: Cash offers are best but 90% of the offers I see as an REO agent have financing. Banks prefer as much downpayment as possible, pre-approval attached to the offer and conventional financing with short loan commitment time. If there are two offers of equal dollar amount, the financing is carefully considered so don't underestimate this!
3). No subject to sale contingencies.
4). Quick closing date.
Good luck on your foreclosure purchase~ !