Generally you'll find residual rates are higher than the car is actually worth (unless you get a really low residual, like 760Li's carry a 48% residual after 36 months). What leasing is, like you said, is paying the depreciation of the car, plus interest. Your stealer is probably going to hit you with terms such as cap cost reduction (which is, in reality, the down payment), residual (what the buyout will be after the lease term), money factor (which if you multiply by 2400 you will get your interest rate percentage), acquisition fee (which is just an added fee), disposition fee (the fee that the company charges to "take your car back"), allowed mileage. The allowed mileage part is pretty self explanatory. If you end up buying your car out at the end of the lease term, there will be no disposition fee. The amount of sales tax you pay is based on your residual.
Say, for example, you decide to lease a 750Li sedan for 36 months. If the MSRP is $88,000, and the residual is 62%, that leaves you with a buyout of $54,560 at the end of the lease term. Your monthly payment will be $33,440 divided by 36 plus interest. Since you most probably will have negative equity in the car for the duration of the lease, make sure your insurance company has GAP insurance. That way, if God forbid your car gets stolen, or totalled, you won't have to pay the difference between your payoff and what your car is worth to the insurance company (which is usually less than your payoff) BMW has an option for multiple security deposits (up to 9), that will lower your money factor. Your cash due at signing is your downpayment, bank fee (acquisition fee), registration and plates fee, and taxes. In some states you can choose to roll the taxes into the payment if you wish.
In regards to yearly mileage allowance; if you choose a lease with 15,000 miles allotted per year for 3 years, and at the end of the lease term you have 50,000 miles on your car, you'll have to pay a penalty (I believe it's 20 cents per mile). However, if you're under your allowed mileage, you cannot get 20 cents per mile back.
Unless you really need to be at a number for your monthly payment, I would not recommend putting too much of a downpayment on a lease; reason being that if your car gets stolen or totalled, your down payment goes down the tubes, you never get that back.
I hope that explains most of it! If I missed anything please let me know
Good luck!!