I think she may have quit laying for a few weeks post treatment with formic acid. Did that trigger supercede?
Yes it can.
Anyway what happens to the queen? Will they keep her until they know they have a healthy new laying queen?
Yes that is what they do. Therre are occasional but rare exceptions.
It seems late in the year for a new queen to get mated. Who kills her?
You mean who kills the old queen? The new queen does, after she is mated. She does not kill her outright as her sting is now used as an ovipositor. Instead she follows the old queen around, biting it and badgering it. The old queen gradually loses it's hair, it's wings, becomes weak and frail, and eventually dies.
Why does she not destroy the supercede cells? Do the workers keep her from doing so?
Yes, the workers prevent her.
Will she fight the emerging queens?
No
I have heard that she can coexist with her daughter in the hive. How long?
Varies but normally weeks to 2 or 3 months.
The old queen dies.
Highly unlikely
Will she swarmext spring?
That depends entirely on the condition of the hive. Bees like all living creatures are driven to reproduce, and any healthy hive given suitable circumstances will try to swarm.
Should I destroy the supercedure cells or let nature take it's course?
There is a small element of risk either way. If the cells hatch, the bees normally protect the queen from the resultant supersedure virgin, until after she is mated. If the virgin fails to mate, the virgin eventually disappears. However there is the odd rare case where the old queen is killed by the virgin prior to her mating. So me anyway, if I think there is a chance the supersedure queen will mate I will leave things. If I know there is no chance she will mate I will destroy the cells, but I don't get too worried if I don't, the huge majority of the time these things work out.
Formic acid does mess the bees up quite a bit, and unseasonal supercedure cells can be built after treatment, because the bees are messed up.