How long do mums continue to flower if I plant them now?

Flowers generally last about two or three weeks, depending on the outdoor temperatures and how far along the blooming process was when the plants were purchased. Flowers don't last as long when it is still hot and daytime highs are in the eighties. Some gardeners use mums as temporary color in the landscape, and when the flowers fade the plants are removed and discarded in the compost pile. Chrysanthemums will, however, bloom in the fall garden for many years if grown properly.

https://homeguides.sfgate.com/long-mums-last-bloom-63422.html

Probably until the first hard frost

Not long.

If they are "hardy" they should come up next year! Usually a few dollars more than other mums!

I have kept my mums in there pots till ready to plant in ground, after the blooms are finished ( around 1st frost). Prune them back about 3 inches from the ground. Then wait for the following year, year after year they great fuller and more beautiful.