Wow 7.75% seems really high??? Is there a reason you want to lease, ie. business? If you want the car and don't need to lease I'd suggest you go to your bank and try to finance the car. I'm sure you can get a better rate.
In terms of trading in your car, as far as I'm aware, you don't have to put the full amount towards your car, you do get a tax break on the amount you do put towards your car though. That amount is not taxed.
A word of advice, I'm sure you're aware of this, but negotiate your trade separately from the total car price. Also, it might not be the smartest thing to tell them, before you agree to the price of your trade, that you don't want to put it all towards your car.