대다 is auxiliary verb for repetition For more auxiliary verbs, refer to https://www.italki.com/question/365005 e.g. 웃어 대다 to keep laughing 졸라 대다 to keep begging 소리를 질러 대다 to keep shouting 놀려 대다 to keep teasing 그러고는 그냥 온갖 소리를 질러 대요. And then he just keeps shouting with all the noise 저를 이리저리 떠밀고 놀려 대고요. And he keeps teasing me while pushing me here and there. The suffix A/V + 고요 adds more information about something mentioned earlier: For example, 여기에서 버스를 타는 것은 편리해요. 값도 싸고요. Travelling by bus is convenient here. And it’s cheap too. 웰빙 음식은 맛있어요. 건강에도 좋고요. Well-being (wholesome) food is delicious. And it’s good for your health. 긴 영화예요. 그리고 정말 지루하고요. This is a long movie. And it’s really boring as well. 그는 부자예요. 인기가 있고요. He’s very rich. And he’s also very popular.

Both 질러 대요 and 놀려 대고요 are based on the verb 대다. 대다 has many meanings, but here it functions as a helper verb in the form of [V + -아/어 대다], to add to the main verb V the sense of doing something repeatedly or without restraint (울어 대다 = to keep crying). So 소리를 질러 대다 means to keep yelling (소리(를) 지르다 = shout, yell), and 놀려 대다 to repeatedly taunt someone (놀리다 = taunt, tease). 놀려 대고요 = 놀려 대다 + -고(verb connective form) + -요 (politeness suffix). -고 more commonly connects two expressions like an "and" (차가 크고 좋다 = The car is big and nice), but it can also come at the end of a sentence to 1) add something to what's already said, or 2) end the sentence with the nuance that one has more to say (it's the #1 usage here). - 그러고는 그냥 온갖 소리를 질러 대요. => Then they just go on yelling all kinds of things. - 저를 이리저리 떠밀고 놀려 대고요. => And they also keep pushing me around and taunting me.