Looks Awesome!!!
The dust and dirt will settle in the pond or get trapped in the bog in the next few days. The only way to prevent that would have be to fill the bog with gravel and water, then pump out the dirty water b4 it gets a chance to enter the pond. Not a big deal, you will see a big difference in 24 hours or so.
I would run the system as long as wheather permits. There will be cold water stains of BB and exzymes that will like the areation and continue to grow in cooler temps.
You could bring the pump inside, just make sure you keep it in a bucket of water so the diaphram doesn't dry out. You can also run it through out the winter by running out to the main pond with a little extra plumbing. I will be turning down my father-in-laws pond soon and using the same method. The pump is plumbed to some 2" trash hose and goes toward the other end of the pond. It has a 90 that is turned towards the surface, it then acts as a bubbler and keeps that end of the pond open and allows for the skimmer to run and collect debris year round. I will gets some pics., but not for another few weeks, or if the wheather gets really cold soon since I'd prefer to work in warmer water.
For draining, i would think all that is needed is to turn off the pump and let it drain back into the pond. That would require popping off the check valve, or just letting it drain down to the spillway level. Any plants in the bog will go dormant and not need to be in water, or ice in this case. The gravel and snow will act as insulation, and winter rains and snow melt will be impossible anyways to drain out. Using natives is the bog is another way to insure they will survive the cold.
I will post some pics. of my plants soon. I forgot to mention Joe Pye Weed. This plants is a North American native and does great in the shallows. I have it planted half in the pond and the other half is going in the surrounding soil. When I bought it, I washed the soil from the roots and litterally sliced the root mass in half. I split it so it is planted in the gravel around the pond and in the pond itself. This thing gets splashed by the falls all summer and is an iceberg in the winter. It gets tall, (5'+), but dwarf varities are avail. and can be used on the edges of the pond, not necessarlily in the bog.