1.) Yes, you should include all of your vehicle expenses incurred for business in 2021. If you don't enter it, you will forego your vehicle expense deduction that helps offset your Dasher income. TurboTax will ask you for the date you purchased it (Jan '20) and they will also ask for the date (May '21) you started using it in business (placed in service date).
2.) Your purchase price is the sales price before down payment/trade-in, plus taxes and fees to purchase. This amount will be used to calculate depreciation. Loan fees and the down payment are factored into the amount financed - this amount is not deductible, but you will be given the opportunity to deduct the interest portion of the payments after you enter your mileage data for 2021.
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