Like Azelor said - long term average 0°C at the pole (and yes it is exactly the pole, e.g Lat 90° and - 90°) is not consistent with an Earth type planet.

It would be possible for a planet much nearer to the sun and/or having a quite powerful green house effect.
As yours is a fantasy planet, you can impose 0° on the poles. You can actually impose anything you want (on Venus metals boil on the poles ).

But then yours would be an extremely hot and very Earth unlike planet. You could surely NOT impose only 25° on the equator. It would be much closer to 60 or even 80°. Water would almost boil.
If you want to have an approximately Earth like planet, best is to fix Tp anywhere between -30° and - 50° and then set Te some 60° to 70° above it.

Why do the precipitations follow approximately a \/\ profile in latittude dependence ? Simple thermodynamics.

At equator the temperature is highest. As long as there is water (one could imagine a planet where there are no oceans on equator), the evaporation is highest. And what evaporates must condense.
So as the hot moist air rises, it expands and cools down. Result are maximum precipitations.

After that the air is moved by the Hadley cell towards the N at high altitude. It rains less and less. Then it starts to go down, compress and warm. Its relative humidity is very low so that precipitations are virtually 0. That happens around 25° N and that's why there are deserts at those latitudes.

Then the next cell will do the same thing farther N. But the difference is that we start relatively dry and we arrive to the temperate regions where temperatures fall so that relative humidity rises. It will rain more but not as much as on equator.

The last leg of the \/\ is quite messy as we approach the pole. The temperatures get extremely cold from what follows that relative humidity is almost 0 and it can't rain (actually snow). Antarctica is by far the driest area on the Earth.