What does the first part of 它一把拉着我跳回了家，我成了兔子，而它成了我的模样 mean?
Reading the story, "它" in "它一把拉着我跳回了家" is referring to the rabbit. --> "It (the rabbit) pulling me along and hopped back home"

As for the function of "一把", you can reference my answer to this question: 


  It is a description of a manner which implies "abruptly with no hesitation"
  There are similar adverbial phrases:
  
  一手 (with one move of one's hand)
  
  一手搶過我手中的香煙 (with one move of his hand, he grabs the cigarette off my hand = he takes the cigarette off my hand with one move of his hand)



它拉着我跳回了家 = he pulling me hopped back home
它(一把)拉着我跳回了家 = he (with one swift grab) pulling me hopped back home



  我成了兔子，而它成了我的模样
  
  I turned into a rabbit, and it (the rabbit) turned into my appearance. 
  
  Notice the phrase "it hopped home", which mean before they got home, the rabbit was still a rabbit.  Once arrived home, the rabbit turned into the child; and the child turned into a rabbit (obviously a dream)