I�ll give this a shot.
1. Is the bottom box always used as a brood chamber.

As far as I know it is. Different beekeepers try different things, but I�ve only seen the bottom box used as a brood chamber.

Do you use it over and over again in the same way? Do the bees just keep hatching and using it again for laying eggs?

In the bottom box (brood chamber), the bees will have brood, honey, and nectar. Mostly brood though. They tend to store honey over brood (higher up in the hive). As the brood emerges, the bees clean out the cell and the queen can lay another egg there. Or if they choose to, they can store honey or nectar there.

Does the queen go back down to it, or is she forever climbing up?

I�ve observed that queens go both up & down. They�ll go where there is available space to lay eggs. In the winter, the queen isn�t doing a lot of egg laying & the cluster generally moves up, into the stored honey, as the winter progresses.

Do they keep using the same foundation--or do I need to replace it? ANd how often?

The bees place wax on foundation and it becomes drawn comb. They�ll use the same comb over & over. Some people recommend changing out old frames of comb every three years or so because the old wax can become contaminated with pesticides, etc. Sometimes I will do this, but I try to hang onto drawn comb as long as possible since it requires energy (time/food) for bees to turn foundation into drawn comb.

2. If you aren't interested in harvesting honey, could you simply have a hive that sits there and where the bees take care of things themselves? Will they eventually outgrow it or abandon it?

They will likely be OK, but will also probably swarm most years. That�s not necessarily a bad thing if you don�t want honey. It helps replenish wild (feral) colonies. In a swarm, enough bees will remain in the hive to build back up to size. Bees don�t always read the instructions though & they�re not totally predictable. You�ll want to keep an eye on them occasionally to check for too many varroa mites or other pests.

3. How much honey does a regular hive need to winter over? I have medium boxes? How many boxes should I have?

That depends on where you live. The longer your winters, the more food storage they�ll need. I�m in the mountains of Tennessee. I started out using double deeps but have found that a deep brood chamber & a medium do fine for over wintering. If you have only mediums, use three. I do feed heavily in the fall to ensure the bees have enough stores for the winter.

4. DO you extract honey from the frames where there was brood? Or do you always have to have the queen extractor on to keep the brood out. Or do the bees clean it up after the brood leaves enough that you can use that honey?

If I find brood in my honey supers, I wait until they�re emerged & then extract the honey. I do own a queen excluder (one), but have never used it. The bees will sufficiently clean out a cell that had brood in it.