At first, he may cope by denying it. He might refuse to believe that Beckett is really dead. He'll keep waiting for her to show up, going to all their favorite places in the hopes of seeing her again. As time passes, though, he'll slowly start to accept the reality and find new ways to keep her memory alive.
He might cope by writing. Castle is a writer, after all. He could pour all his emotions into a new book, using it as an outlet for his grief. The book could be about their relationship, or a fictionalized version of the events leading up to her 'death'.
One possible storyline could be that Castle goes into a deep depression. He might isolate himself from his friends and family, constantly reminiscing about their time together. He could also become obsessed with finding out the truth behind her 'death', suspecting foul play and starting his own investigation, even if the official version says it was an accident.
He might be in denial at first. He could keep trying to convince himself that there's no way Beckett would leave him.
She is probably angry and frustrated. She would demand Castle to explain why he thinks so and would be hurt that he doesn't trust her.
She could be defensive of Beckett. Since Alexis probably has a good relationship with Beckett, she may try to convince Castle that he's wrong and that Beckett didn't mean any harm. For example, she might tell Castle about all the positive things Beckett has done for her.
He might become more protective. He'll be constantly on guard to make sure nothing bad happens to Beckett.
Well, first, give yourself time to grieve. It's okay to feel all kinds of emotions. Maybe join a support group to talk to others who have gone through the same. And don't be afraid to seek professional help if needed.
He might react with jealousy. He could glare at Josh and then give Beckett a hurt look before storming off.
There could be. The Castle fandom is quite creative, and a pregnancy plot for Beckett is a popular trope in fanfiction. Some fanfics might focus on the romantic aspect of it, with Castle being over - the - moon and taking extra care of Beckett. Others might explore the more practical and challenging sides, such as Beckett still trying to solve cases while dealing with pregnancy symptoms.
Maybe he saw Beckett spending a lot of time with another man without knowing it was for a case. Castle is a bit of a jealous type, and he might overreact to normal interactions.
It could be that Castle is overprotective. He loves Alexis so much that even the slightest sign of Beckett not being on the same page as Alexis in Castle's eyes makes him think Beckett hurt her. Maybe Beckett gave Alexis some career advice that Castle thought was too pushy.