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It would be impossible to know wether or not the finance charge's are $30 for that date range without more information. Such as APR, amounted financed, finance term (54/60/72 mo?) and day's until first payment. An easy way to find out what you will pay in interest in call Chase and get a payoff amount for 10 days out, and also ask for the Per Diem. The Per Diem is the daily amount of interest charged on a loan.
You will abosolutely pay finance charges for the days since you contracted until you pay it off. Paying it off won't necessarily hurt your score unless its the only installment loan you have open.
As far as the extended warranty it just depend's what it cover's, deductible, mileage/term expiration. Always know that extended warranties can be negotiated down in price, that warranty could very well have $500-1000 worth of profit in that price so try and negotiate a better price once you find out the terms of the warranty. Is it a comprehensive ( bumper to bumper) or a power train warranty?
With my rough calculations your going to pay north of $100 in finance charges. Unfortunately it's your responsibility to pay those charges regardless of what the dealership told you.
A powertrain warranty extending it 40k miles for $1400 sounds a little bit pricey, if that is something you are interested in, I would recommend trying to negotiate the price down to 900-1000. But, do know, that a powertrain warranty isn't all that it is cracked up to be, they normally only cover major malfunction's within the engine/transmission/driveaxe/awd four wheel drive system. For example, if a check engine light comes on for a mass air flow sensor that is faulty, that is not covered under powertrain warranty. Now if your transmission goes out or start's slipping, that is covered. If you are interested in a warranty I would inquire about a comprehensive warranty that extends it from the factory 3/36k to something higher but only at a decent cost. A comprehensive warranty will cover nearly all items on a vehicle excluding routine maintnance item's such as tires, belts, hoses, etc.