I will do the best I can to answer you, but I haven’t been to very many funerals in the last several years.
From what I know someone in the family contacts the parish at which they wish to have the funeral and make all the arrangements with the appropriate person there. Sometimes it is the pastor, sometimes it is a different person. It all depends parish to parish.
Now, I believe a funeral mass is much like a regular mass, but more focused on the person that died and time for eulogies and such. Again, you’d have to talk to your local parish to find out what is done there.