A: Mums in containers are beautiful but need to be cared for properly.
When mums are in bloom in containers, try to water the pots from the sides of the container as much as possible. Open Mum blooms give up quickly when they stay wet too much. Light, natural rainfall is fine, but if storms are in the forecast for several days, tuck the pots up under your patio. Closed buds are safe; this care is only necessary when the plants are in open bloom.
As for pieces dying out, mums are very top-heavy in containers, with brittle stems prone to breakage. Handle mums by lifting by the containers, and while watering, use a soft water flow instead of a forceful one. Direct water hose flow can crack mum stems apart.
Neither of the problems you have are pest-related. Mums need careful handling to do their best!