They have scarcely gone out since the baby was born.
This one is tricky because the natural temptation is read seit as a proposition and think that its object must be dem Baby, but in fact it is acting here as a conjunction with no implications for case. Waehrend does the same thing to me often. Grrrr. 
Maybe I’m blind to your problem because I’m a German native speaker and am used to it, but German “seit” is no different than English “since”.
These sentences are in and of itself complete sentences (and the German does not have a Dativ):
Therefore, “das Baby”, not “dem Baby”.
The following sentences however use the Dativ.
Other than "Seit das Baby geboren ist“, “Seit dem Regen” is not a “complete sentence”.