Welcome to Beemaster, heartfarm! What an amazing way to get your first bees!
All of that background to ask-- how long do I have until the hive is "settled" and have to take strong considerations for relocating the hive?
If they have been in there for a week, they are basically already settled. The general rule of thumb for moving hives is "3 feet or 3 miles". Bees visually memorize the location of the hive when they are first learning to forage, but once they have done so, they will leave the hive taking for granted that their memorized location is accurate. (You wouldn't expect your house to move either!
) The bees won't have trouble finding the new hive location if it's moved about 3 feet or less, but any farther away than that and the bees won't be able to find it reliably. However, if you lock the bees in overnight, so none of the foragers get left behind, and move the hive several miles, everything will look so different that the bees will memorize the new location. So unless your property is VERY large, moving them to another location will probably be a gradual process. There are some ways to try and get bees to reorient to a new location, like putting a branch in front of the entrance to confuse them, the success of which is variable based on what I've heard, but I've never tried it myself.
My plan is to move the frames they are building on (now 7 days) into this "new" hive being delivered after they've been in there for about 3 weeks total. The new hive is already painted white and may fare better in weather (sun, monsoons, some light snowfall). Is that wrong?
White is a great color for a hive, and unless you lived in an extremely cold climate it would not be a problem (and honestly, I've seen plenty of beekeepers in Alaska and far northern Canada with white hives too). I have hives in all colors. The only thing I'd be concerned about would be black.
Do I need to, or is it better to keep them in the existing super (proper super just not painted) and just set that on the new bottom board/stand, and add the medium on top of that?
It should be easy to transfer the frames into the painted deep when it arrives if that is what you prefer. I'm assuming your climate is very dry, so there would also be likely no issues using that unpainted box for a season. Even in my extremely humid climate, I have a friend who doesn't paint his boxes, and they hold up okay. If you'd rather have them in the painted box though, you can just transfer the frames they are on to the new hive on the new stand, and I doubt they will have any serious trouble as long as the location remains the same. The difference in the height of the hive could confuse them for a few hours, but they should get the hang of the new entrance quickly enough.
I ordered a kit from Dadant that includes a 10-frame deep super and medium shallow. I'm waiting for the hive stand and queen excluder. Right now I have them in a clean "temporary" deep super I had in the barn, and am still looking for the best location. We are rural, 5000ft high desert and sometimes winds of 35mph so am considering building an adobe wall long term for windbreak. I have pasture and orchard 0.1 miles from the house but lots of variables with cows, irrigation and critters so am considering ample space uphill nearer the house. Right now I have the hive set on boards with cinder blocks just inches off the ground (all I had available) just outside our courtyard wall under a mesquite bush with weight on top of the cover.
Just to put your mind at ease, nothing about this setup sounds problematic to me in the least. I keep my hives on cinderblocks permanently (I use 8 frames, so they get tall fast). Also, just a quick correction on terminology. There are 3 Langstroth box depths: deep, medium, and shallow. A super is a box of any size placed over the brood nest as space for the bees to store honey.
Congrats on becoming a beekeeper! Please don't hesitate to ask us any and all questions you may have. We never get tired of talking about bees here on Beemaster.