1. Was it stuck inside of her? Is that even possible.
"stuck kits" are VERY common in dwarf and small breeds - just do a search of this site
2. Is there more that will be coming (alive or dead)?
There could be more, likely dead if they have been in there too long, but possibly alive.
3. If she has more will she know to have them in the nest box?
maybe, maybe not. It is very traumatizing for a doe to pass a stuck kit so they are not thinking rationally. As long as their is more than 1 born alive and they huddle togeather they should do OK out of the fur nest for a few hours.
4. Was it the mating yesterday that killed it, or caused it to come out last night?
The mating DID NOT kill it but helped to increase her hormones and hence uterine motility so she could pass it.
5. Do we consider her pregnant from the 1 cover on Saturday, or the 5 covers yesterday (Monday), and start counting 31 days?
It is highly unlikely that the sperm got around the kits(s) in the uterus to fertilize the next batch of eggs in the Fallopian tubes so I doubt she got pregnant from those breedings. But keep an eye on her in 30 days just to be sure
I wait at least a week before re-breeding a doe who had a stuck kit so she has time to heal up internally and to rule out her having an infection. Stuck kits can cause tearing of the reproductive tract and bacteria can invade so please monitor her behaviour, eating, drinking and bathroom habits to ensure she is functioning normally.
If they have a second problem pregnancy they are removed from my breeding program as I do not want her sons or daughters passing on these defective genes.