Payments is a more complicated way of looking at your financial outlay. Looking only at payments casts confusion on the total cost of your car. For example, if I had over-paid $500 for my car, my montly payments would only go up by $9 (60 months at 3.9%). On the other hand, if you negotiate the best possible price for your car and trade, your monthly payments will be the lowest possible under any combination of payment terms. If you understand this point, you would never negotiate on the basis of payments. The fact that my dealer constantly tried to show me my payments goes to show just how dangerous negotiating on the basis of payments can be to those who don't understand it.
While negotiating on payments can become confusing and dangerous for the buyer, it isn't much different to the dealer than if you had bargained on the price of a car and trade. They are going to calculate the value and find out what those payments work out to. If they are making enough money, they'll sell it to you. If not, they won't. If it is hard for them to give you fair value for your trade, then talking about payments doesn't change that fact. You may not be talking about bottom-line prices with your salesperson, but that's how the floor manager is going to look at your offer.
If you know what your car is worth, then you know how much to ask for it. If you know how much the new car can be bought for, then you know how much to offer. There is no confusion. And if you don't want to hassle with bargaining on your trade, take it to CarMax like I did, where I promptly got full Kelly Blue Book trade-in for my '92 Accord. With a 7-day 300-mile quote, I took that offer to the Mazda dealer who couldn't beat it. Not forcing the Mazda dealer to give me a good trade-in value means they have more room to work with on the price of the new car. It's a win-win-win for all three parties.
Your APR is determined by your credit score and downpayment, nothing else. The dealer should be able to offer you the lowest possible APR no matter how the rest of the deal is structured. It's not even a bargaining point.
The rebates are from Mazda to the buyer. The dealer acts as a middle-agent. The dealer gives the rebate to the buyer and then gets reimbursed from Mazda. There is no "signing over" of anything. If you do not mention the rebates until you've arrived on a final sales price, you get the rebates as a further discount on the price, regardless of what your negotiated price is, even if it is below invoice.
BTW, I bought my Mazda3 for $118 below invoice, that's before $1000 in rebates (both owner loyalty and college grad). I also got 3.9% financing for 60 months. If you want to think about this in terms of payments, I started the week out with a '92 Accord EX, and ended the week with the Mazda3 in my signature and a payment of $296 for 60 months (estimated as if I had put the $1700 I got for my old car in as downpayment.)
Out of curiosity, what kind of deal did you manage to get?