Those are fungus gnats. They are harmless to you, but they indicate that there is either insufficient drainage, or you are watering too much. They will also migrate to other plants and can carry viruses that may damage your collection. You said you only water once a week, then the water isn't draining well, the soil is breaking down and growing fungus, which attracts the gnats. They lay eggs, die, and the process repeats itself over and over again every 1-2 weeks.

Your plant is also suffering from what looks like root rot, again from being too wet. Don't let water stand in the tray under the pot. Let it drain completely outside for an hour or two before putting it into the tray. It may not recover from the rot, but its worth a try.

First off, completely replace your soil. I recommend using Kellogg's Patio mix. Good stuff and well draining. Lightly brush off the remaining soil from the roots, bag up and throw away all of the old soil, and then dust the roots with some powdered cinnamon, which will help inhibit further fungal growth. You might also think about moving up to a bit larger pot. If not, be sure to scrub out the old pot before putting the plant back in. Wait at least 2 days before watering the plant, then wetting it just until water runs out the hole in the bottom. Let the top 1-2" of soil get completely dry before watering it again.

Also, a north window is not enough light, which is also contributing to the gnat/rot problem. Your banana needs as much light as you can get it. You say you live in SoCal, so you might try growing it outdoors, gradually moving it from light shade to dappled sunlight to full sun over a period of 2-3 weeks. Bananas love full sunlight and ample moisture, but also do not like to sit in wet soil. If it's outside you can forget the tray.

If you can't put it outside, try an east or south facing window. If it's a southern exposure, be sure not to let the plant get too close to the window or you will burn the leaves.

Good luck!

That was an excellent follow up !

You rock !