I was finally able to verify the whereabouts of my mother's Busan home with her oldest sibling just a few minutes ago (along with many childhood memories).
Apparently, my mother's memories were so faint, due to her very young age at the time, that I did not even explore the correct neighborhood! My mother was right in that from Seomyeon and its Lotte Department Store (which, as it turns out, was a high school back then, commandeered by the UN as a military hospital during the war), a thoroughfare led to a railroad bridge, and her place was nearby. However, we had picked the wrong thoroughfare and the wrong railroad bridge; no wonder my mother couldn't remember a thing!
I was also told that the house still stands today, if only because it sits on Korail land and the government won't give building permits to the residents for a teardown and rebuild. The neighboring houses, sitting on private land, have all been torn down and rebuilt into something nicer and more modern ever since.
I have no plans to return to Busan during my current South Korean stay. However, when I come back in the future, I'll make sure to return. I'll also visit the site of my father's teenage home at that time.