To get them feeding its best to have the feeder under a bright light preferably one which has UV and gives of a bit of heat, maybe a 30w-50w reptile basking light? they should feed for about 30 seconds to a minuet, 2 to 3 times a day, unfurl the proboscis and place it in the nectar you should see the end moving about if they are feeding. If they refuse to feed it might be best to change the concentration of the nectar, as if it is too dilute or too strong they refuse it as well.
It's best to not put salt in with the nectar as it can cause them to dehydrate if they drink it all the time, you can try putting some in a small dish with a wet towel on the bottom of the enclosure. Do you use kitchen salt or rock salt, the butterflies prefer to have a mix of minerals, so it might be best to use Himalayan rock salt if you can get it easily, some supermarkets have it.
Quite a few African papilio species diapause, as for how long it will last, I'm unsure, I've had some that go for 5 months before hatching out, but you might be able to force it out of diapause by raising the temperature and humidity though. Or I've heard that some papilio respond to light for coming out of diapause although i haven't tried it.
I'm not sure on the exact species, but if the young bark is very fibrous and aromatic it fits all the criteria in leaf and branch structure to be a Annonaceae.
If it isn't though you can always use magnolias as I've heard that they can also work, although I've not tried them personally.
Hope this helps,
I've noticed it with monarch and most nymphalidae, although it usually only happens to freshly emerged butterflies. The way you are doing it shouldn't damage the wings though.
How do you hold the butterflies after you've caught them? I've noticed that some species if you pinch the wings together you sometimes brake the wing where they connect to the body, with the ones that flap on the floor does one side seem a bit limp? if so the tend to bleed out quite fast.
The monarchs and swallowtails should be able to live for about 2-3 weeks some times up to 6 weeks.
How do you mix up your sugar solution? i find about 15-20% sugar works best and with swallowtails I've noticed they can't have any sugar that has started to ferment which usually happens after about 7 hours (faster if you use honey.)
As for misting them i find that soaking the whole cage once a day works best, or you could just spray the proboscis a couple of times a day. It might also be worth putting in a shallow tray of wet sand with a bit of salt in for the swallowtails.
If i had to guess i would say over heating or dehydration, do you have any shade in there?
Hope this helps