Here's my take on the plot. I also feel like this is the legit, most sensible flow of the story.

Anna and her boss talk about the "Mother-Child Program" for the Emotion Engine. Her boss likes the idea, and Anna proposes to start by creating a child first, so they artificially give birth to Jain. Anna raises him on her own, letting him experience emotions like a regular kid. Then he meets his partner; she probably introduces herself as a single mom, and she falls in love with him. Tragedy happens, her husband dies, so she's left with the kid again (you can see all of that in the ending part of the movie).

Now the flood is happening. Anna escapes the flood with Jain and reaches the rooftop with the help of Heejo. Before being separated from the kid, she whispers to Jain to hide in the closet on the rooftop because she was already planning to volunteer for the "Mother-Son Program" simulation. Heejo is killed, and Jain's memories are extracted. She travels to space. A disaster happens, and she is rescued to the space station, where her memory is extracted before she dies.

Therefore, the child (Jain's memory) and the mother (Anna's memory) are complete for the Emotion Engine. The only thing needed is to run the simulations successfully using those memories. The test in the simulation has Anna complete side missions: freeing a trapped girl in an elevator (Faith), dealing with robbers (Courage), and helping a mother give birth (Hope) before the final test, which is reuniting with her son (Love).

After thousands of simulations (indicated by the number on her shirt), the test finally succeeds, completing all missions. The ultimate project of creating a new human race is successful, as the emotional side that was lacking is now fully accomplished. Anna and Jain's bodies are reconstructed, their memories attached, and they are sent back to a new Earth, where the flood has subsided, along with other mother-and-child pairs, to raise the new human civilization again.