You have to understand, sister, that no Imam likes to process a marriage that a major component of the conditions of marriage is missing. In most of such cases, the judge or the Imam will have to communicate with the father of the girl to convince him and if he sees that the father is being unfair and lots of Haram might be practiced he might apply some views of Fiqh to process the marriage under his authority.

So, my suggestion is to talk to the local judge or Imam in the Islamic court where you live and have the Imam speak to your dad about this situation.

Here in the US, that’s what we do when we face such case like yours and in many cases it might work and some cases it might not and you need to be ready for the worst scenario.

As to your specific questions:

For couples marrying for the first time, I do not see the permissibility for a private Nikah at all especially in your case where the father is not in approval. Private Nikah is usually practiced with all requirements like Wali, two witnesses, Mahr and its kept private mainly from another wife so it might not break the first family. I did not see that in your case.

Islam gave both daughter and father the veto right to block people unwanted in their life. You can reject anyone your dad imposes on you and in the same way your dad can reject anyone becoming his son in law. Both daughter and father are asked to be fair but if father insists, he can still block this marriage and you should be ready for that.

In most cases, being persistent on your request from dad for a long time, having people involved and getting support from a family member beside you will eventually make the father change his mind insha Allah.

The brother cannot be your Wali with the presence of your father. Only by permission of father, a brother can be the Wali.

Now, if your brother wants to be responsible to take that risk then the marriage must be done at an Islamic court and the Judge must rule based on the rejection of the father as non-binding ethically and that the bride is mature and her Wali is available, it might work.

Allah Almighty knows best.