Are caterpillars during the pupa stage?

No. Insect (and thereby butterfly) development knows four stages; egg, larvae, pupa, adult. The caterpillar is the butterfly's larva. The pupa is the chrysalis, the immobile hanging thing in which the larva (caterpillar) changes into the winged adult (butterfly). ^^

All metamorphing insects do the same, including for example flies (egg, maggot, pupa, fly) and bees (egg, larva, pupa, adult, but in the hive so you don't see!).