TM,
I'm sure the experts will come along and correct anything I mess up, but here's the gist (and I'm assuming some sort of winter/dormancy even there where you live. Take what I'm saying with a bit of salt because your conditions will be different, or at least less lengthy).
1. I cut my bog back to the ground, though some sites say to leave the vegetation to prevent fungus from starting in plants that are reedy (and therefore hollow when they die back). For me, it's just like my flower gardens and I cut back and clean up. No real sense, imo, to leaving debris to add to any possible decay and therefore gas buildup. When I turn off my pump for the winter, my bog water level drops and really, there is no connection, but there's also no sense in NOT doing it per se. So, I cut back and clean up. From the reading, the beneficial bacteria will be inactive as the temps drop. Here's the difference (maybe) from your setup to mine, as for sure, temps are going to drop way below bacteria activity levels. I forget the exact temp, (40 F sounds right), but if your winters are less than that, your bog won't be functioning as it could, so not a worry anymore.
2. Think I answered this one; I turn mine off. The reason I pump to the bog is for filtering and if there's no active plants, and no active bacteria in the peagravel below, there's no purpose for pumping water to it. At least, re filtering. Now, some will say they let their pump run the whole time and with good effect, and with good purpose. For me, there is no gain, so I turn mine off, keeping it in the deep end and far below ice level. I don't like the idea of mixing the cold water on top with any possible warm layer stratification below. Just my take on the whole pump on/off in the winter. YMMV
3. Re the pump; again, I think I answered how I do it and why. I'd not have any water moving in places that can ice up. You might cause a crack in the spitter and falls might ice over, forcing new paths for the running water to find. Just my take. I figure; the fish are going dormant, the humans are definitely going dormant, so why? No need to create more aeration--the fish don't need it in the dormancy/cold water capacity to hold oxygen. But some do. I'd just worry about that one time something ices up and a problem occurs. For me, not worth it without obvious (to me) gain re the pond/fish. Nature doesn't do this (for still ponds), so I figure I'm modeling what I see. The only thing I do is make sure the ice cover doesn't last too long, and ideally, I keep a hole open the whole winter (by having an airstone running. There IS an added benefit that should oxygen levels drop, I'm helping replace what is lost). For you, with milder winters, if you have little ice, little time of ice cover, and your pond is not a cesspool of rotting debris with an overpopulation problem, well, I'd just let everything be. If your fish are active and you don't worry about freezing, I'd run the pump. If the fish do go dormant (not the right term, but I can't remember the correct one at the moment--heh, more of that crs you were asking about in another thread!), the need for filtering is lessened a lot, imo.
3.5. I don't test the water, but it doesn't hurt. I tend to watch how everybody is doing and if I see signs of distress in anything living in the pond, begin to analyze the conditions. If your fish are not eating (much; they'll nibble on whatever algae/plants you have under the surface if they're hungry in the winter), and you're not feeding them, their contribution to the waste cycle will plummet, so less chance of any buildup of ammonia, nitrites/nitrates etc. Not much 'living' means not much change, so what's to test, imo?
Anyway, that's how I see it with other ideas having value. Again, really does depend on fish load relative to pond size, ice cover length, and possible damage due to freeze-up.
Oh, and like you, MY remote thermal probe also just arrived and I'll be with you--monitoring the bottom/top temps. I read that unless your pond is more than 7' deep, stratification doesn't occur, though of course, the colder water (and ice) WILL be at the top, but I'm not so sure I'll see 39 F on the bottom as January hits and typically single digits comes to Michigan. For you, I doubt you'll have any issue re this at all; just too dang warm down where you live, imo!
Michael