I don't have koi and was hoping someone with koi experience would address your questions. Most of us here probably don't have experience with natural streams either. However I believe it is safe to say that filtration and other bio-cleaning should not be necessary in a free running situation.

I have a long artificial stream, and (as I have mentioned even recently), I believe it is doing almost all of my cleaning and treatment. I always have clear water unless I have disturbed the stream or pond bottoms; and they become clear within a couple hours or a day depending on the masses of muddy bottom I may have disturbed.

Therefore, I suspect that you probably don't need to do anything. You can certainly remove the leaves and some of the gunk if you like. I do that with the stream two or three times a year. I have only once done a major clean out of the pond and occasionally have removed some gunk with bottom plant cleanup. My pond is six years old. I also have a non-maintained pond that has goldfish that grow better than in my stream/pond. However, no flow, no cleaning, may not work with koi instead!?

If you have a pond with a natural stream I don't think you need filtration and aeration except during the time you might not have the natural flow of water. However,your big question may be about wanting to stock koi. If you have a natural pond, with stream (water flowing in and out), you would probably want to check with your county extension service to see what is allowed by law to stock. Koi may not be on that list. As far as leaves go, I would get them out if possible.