Killing the cat feels like something out of the ordinary. Maybe (see below).
Otherwise, many dogs (not all) would kill a wild rabbit given the chance - most wouldn't catch one.
Reacting to children being born is really common, too. From the dog's pov life just changed, got a lot worse, a lot more boring and frustrating and (often, but again not always) every time they express their discomfort, they get told off. That makes the baby BAD NEWS in their mind.
It can be so stressful for some dogs that the other behaviours may be a side effect of this one. i.e. a very bored dog, not used to the lower levels of attention and stimulation, may attack a cat out of sheer boredom- as an outlet for instinct that they are no longer getting elsewhere.
A vet check and a behaviourist seem the best options. In reality, the only way for anyone to undersand what is happening is to actually see and meet the dog and spend time understanding their typical day and environment.
That is expensive (a behaviourist will cost £200-400) and time consuming. Dog behaviour change is rarely quick; it is achieved very slowly, through repetition and by degrees. The reality is that you may not have the time or energy or money for that.
In which case, rehoming is your next best option.
I doubt you have done anything really wrong here - but this scenario is as old as time and it highlights how having babies can impact (some) dogs and why so many people suggest waiting until children are here and older before getting a dog. That's not meant as a lecture to you, OP. That' meant as a call out for anyone thinking about entering a similar situation, e.g. getting a dog because they are a young couple not yet ready for a baby.