For the coupon issue - I've definitely found that Hertz sometimes displays vastly different pricing when applying a coupon. It does seem very shady but seems to be how their system works.
For the dates - yes, different dates can give substantially different prices, particularly when it pushes you into extremely high demand times like Christmas, I guess New Years is one of those.
You might be able to work around it by booking to 12/29 and then just keeping the car another day, the late fee is only $12
