When Kazuma offered to fly me down to the surface on his dragon, I declined, and instead confronted the doctor to save Sue & company. I ended up fighting through four phases of the boss. I missed the fifth, though, because when Curly offered to trade her Machine Gun for my Polar Star, I accepted... and thus she drowned when I confronted the core.

That's not why she drowned. When there (prior to the boss battle, I think), you need to pick up a tow rope (you'll find it amongst the robot pieces at the bottom of the room where you fight the core). Then when she's drowning you can attach her to you. Then when going through the rapids, you should see a small cabin/room that you can go into. Go in there and put her on the bed. This is as far as I did last time, but there's more! There's a book shelf in there with information on how to drain water from robots. Follow its instructions.

Also, you need to get the booster 2.0, which you can only get if you don't go down to the professor guy when he falls past you (earlier on, I think). If you time it right, you can jump across the gap without the need for a booster.

Or you use to machine-gun to "fly".

Two more hints about what to do with Curly: