Does the town still have birth records dating back to the late 1800s?

Answer 1

Some towns do. Civil registration of births is an idea that didn't gain popularity until the very end of the 1800s. Large cities have some records going back to then, but many births were not registered. It wasn't until the early 1900s that mostly all births were registered. Prior to that, the best records were kept by churches. Baptismal records are often the best available source of birth data prior to 1900.