Unfortunately, you can’t start romances in the post-game. I might change this someday, but all the romances have some cutoff in the story where they stop being accessible. Here is the wiki page on romances..
There are several endings. I’d say about six, but it depends on how much variation you’d consider to be a totally new ending. There is also significant branching through the second half of the game, with most of the events of chapters 4 and 5 being totally variable based on earlier choices.