I read that Beckett had Castle’s baby after she got shot. I don’t recall ever seeing this and I’ve watched all seasons more than once. What’s the deal?

After which shooting?? Kate was shot in the during season 3 finale by a sniper. Castle and Kate were not a couple at that point. Season 8, episode 22 ( the last show ) Castle and Kate were both shot and then they did a 7 year time jump to give us the impression that they had 3 children together. Hopefully this answers your question.