Mileage is a tax time election. You track business miles, dates to and from, where and why, as well as the starting ending odometer reading for the tax year. During the year you book the actual expenses of operating the car, fuel, oil changes, tires,etc
Then at tax time you elect to use the standard rate for mileage as an expense or not. If you make that election then you can not include all vehicle related expenses (depreciation,fuel, tires, oil changes, registration, etc etc) too
The lease payment is an expense usually. If the lease is lease to buy, you need to talk to a tax accountant about how to book that aspect
Yes you keep a mileage record for any vehicle you use for business, it should be separate and contain all the data I mentioned for the main one.