"Baby" without an article is used a great deal by doctors, nurses and midwives to refer to the specific baby - or, rather, foetus - of the woman they are dealing with at the moment. It is effectively used as the name of the infant, so "Baby" would be more appropriate transcription than "baby". "Doctor" in the context works similarly, referring to the specific doctor in charge, almost as a proper name. It's something specific to that sort of situation rather than period or class - "mother" is sometimes used similarly (as in "Has mother complained of any pain, nurse?")

No idea why; it's just the way they work.

"The wife" is definitely more informal usage, suitable for more colloquial settings, and often used slightly ironically these days. Only a very newly-wed man would draw attention to the fact by saying "my wife" proudly!