I'm sorry if it's complicated to answer, but I never understood very well what happened to the bunny that John had. how was it created? who gave it to who? etc, etc. would you mind explaining or sending a link of the explanation?

Okay this is a long story. 

PRE - SCRATCH

The bunny came from the movie Con Air, Dave got it for John on his birthday. John, in a Con Air reenactment gives the bunny to baby Rose in the ectobiology lab and she is sent with it on the meteor. 

Rose eventually repairs the bunny with wool and was going to give it to John for his birthday but John found it in her room and took it. 

He would later give it to baby Jade in yet another Con Air reenactment and she was sent with it on her meteor. 

Jade with her penpal Jake upgraded the bunny with mechanical parts and weapons and gave it to John for his birthday and he named it Liv Tyler.

Liv Tyler would eventually stop the Tumor from being stolen and bring it to Dave and Rose.  He was with them when it exploded and he died, rip in peace.

POST - SCRATCH

The bunny Rose was sent with was given to Roxy who put it in a wizard costume and gave it to Jane for her 13th birthday

The bunny Jade was sent with was given to Jake who put it in an Indian Jones costume and gave it to Jane for her 13th birthday.

Technically those were all the same bunny.

Now the bunny from the post-scratch version of Con Air was in Dirk’s Bros museum.  Dirk stole it and built Lil’ Seb around it and gave him to Jane.  Lil’ Seb eventually got knocked into ocean on Earth where he stayed until he was found by Gamzee and given to Caliborn.