What do you think about my relationship with my father after his death? Is it permissible to agree to an informal division of the inheritance? What about the zakat and taxes on the deceased's money?

All praise is due to Allah for guiding you to honor your father. There is no harm in the form of supplication you mentioned. There is no harm for someone other than the Imam to particularize themselves or others in supplication. Repentance erases what came before it. As for the distribution of inheritance, if it was done with the mutual consent of the heirs and the sister's sound judgment, then there is no harm in it. It is preferable to refer to the Sharia court to identify the rightful heirs and resolve any disputes. As for customary councils, if they rule according to Sharia without bias, then there is no harm in referring to them. Regarding the payment of Zakat from the inheritance, it must be paid for the two years if your father was negligent in paying it; otherwise, it is not obligatory for you based on mere doubt. As for taxes, if they are legitimate, they are a debt that must be paid; otherwise, they are not obligatory for you.