In the present perfect, we use the present simple form of the next verb after forming the present simple "have/had" + participle verb. Other examples would be, "I've heard him speak harshly to people" or "I've watched her cook Korean food." You are correct with the second part of your question: it should be "I got punished pretty badly" (because "badly" is an adverb that tells us how s/he got punished).