Human birth is still a natural and common occurrence, so humans can still give birth easily. The big change in the game was the fact that a replicant, an artificial human being, could give child birth. It's the ethical question that the police chief talks about. The only thing that really separates a human from a replicant is that replicants are manufactured. If they can procreate, what really makes them different from us?
The factory was an orphanage, and it was where the child who was born of a replicant was hidden. To hide her identity further, they faked her death and created a male replicant (K) with the exact same DNA and implanted him with memories of his childhood there. As the police chief explained, implanting childhood memories into a replicant is common because it allows them to develop more contextual insight since a memory is a feeling and otherwise they'd have a computer's logic (no interpretation skills).
The movie doesn't explicitly state it but it's highly suggested that K recognized the place he killed the big replicant (Bautista) from his installed memories, and even as he didn't realize it, felt something was special about the place.
The interpretation of K's relationship with Joi is entirely up to you. As you said, it didn't feel like a lie and that's the point. He loved an artificial being (but he too of course, was an artificial being). What really separates him from her other than the fact he's tangible? She might've loved him simply because she was programmed to, but that means her programmed love for him was completely real. And if he, somehow, felt actual love for her, how is it not real love? The entire point of their relationship is to make you question if something artificial can still be real.
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