this sounds very like a real Irish funeral, very similar to what I have experienced all my life.
In Ireland, it is usually a 2 day "event" an open casket viewing, then prayers as the casket is closed and brought to the church. At the Church, those who didnt go to the viewing usually pay their respects to the family after the prayers. Then the next day, usually in the morning there is a Church Funeral Service and the coffin is brought to the cemetery for burial or cremation. There is usually a procession of funeral attendees to the cemetery. If it is near, less than a mile, usually in rural areas, the family and most of the funeral attendees will walk behind the hearse from the Church to the cemetery. In urban/ city areas, there will be a slow moving procession of cars behind the hearse. Its a mark of respect to the family and deceased that the funeral procession is allowed to pass through traffic.
After the burial, the family arranges food for the funeral attendees. Usually it is the immediate family and close friends / neighbours of the deceased. Depending on the family, they usually arrange food at a local hotel or restaurant. If the deceased was a member of a private sports club or similar type of club, the food is usually arranged for there.
The family usually know an approx number of who will join them for food. It is seen as a mark of respect to join the family, either to mourn the deceased or to support the family. Some people will automatically know they are included and are expected for the food, some people the family will specifically invite to the food.
Most times once the food has been eaten, people start drifting away. Depending on the family, they usually either invite people back to the main family home or stay for the rest of the evening in a bar. It might seem odd to Americans, but to Irish people, its celebrating and honouring the deceased person. As the drink flows, the deceased person comes back to life through songs and music, stories and memories. I will always cherish the stories shared by my dads friends and colleagues the night of his burial.
The cost of the food after the burial is seen as part of the funeral expenses.