I'm concerned that a custom order will take a while, even if I want something very simple. Right now my basement has no exterior door, so I've boarded up the door in my kitchen that leads down to the basement. The basement is unfinished and empty, but I still want to get a door on it ASAP.

Frank Lumber and Door on 15th Ave NE in Shoreline would be my recommendation.

If I am understanding you correctly, you are measuring the opening of the casing. The nominal width of a 24" door is typically closer to 23 3/4" to 23 7/8", I say typically because doors at Home Depot would be 24" but doors at Lowes would be 23 3/4", I was told the reason being that people so often fail to allow for room for the door to close and open that they simply make them smaller at the factory than they say, and they have been doing that for years. But, any interior door can be cut down by a 1/2" (including the ones at the big box stores). The problem is that you are going to have to buy the door and trim it down and then mortise out the hinges, and drill the doorknob. A custom door shop isn't going to custom make this door, they are going to take a standard door and plane down a quarter inch on each side. Now if you can get away with removing less you may not have to deal with mortises. There is usually a little play in the doorknob and you can get away with just trimming one side. If you plan on doing multiple doors it might be worth the trouble to hire someone. Doors are not really that fun.

Not sure if you've found the answer to your problem yet, but having dealt with rugs for a few decades now, the larger odd sizes are becoming more difficult to find. We have to do a lot of our larger sizes (bigger than 10x13) as custom rugs. I'd say look around and see if you can find something ready to go in that size, if not, depending on the type of material you choose for the custom rug, custom rugs may be your best bet!