Recalculation of building insurance and property tax

Settle property tax in writing + by bank transfer; as proof, in case the municipality demands again? The municipalities in Germany are almost 50% bankrupt anyway - you have to protect yourself, otherwise double collection

Such nonsense. What you transfer to someone with the reference "property tax" is totally irrelevant to the municipality. For them, the above-mentioned law applies. And it clearly regulates that the owner must bear the full property tax as of January 1st.

Otherwise, I will in the future transfer an advance income tax payment to my wife and then claim before the tax office, "I have paid after all."

what is agreed upon in the purchase contract? under private law, it is definitely possible to agree on the assumption of these costs...

Yes, that definitely has to be regulated in the purchase contract. That will actually be discussed at the latest during the notary appointment!