While your situation is complicated and difficult, it is great to see how far you have come with your and your partner kids.
I have the following suggestions:
Consistent Response to Child's Action
Child should always receive the same message of approval or disapproval for a given action/behavior from all care takers. For example, if a child wants to eat a bag of candies, both parents should agree on whether that will be allowed or not. The child should not be able to 'test' his/her mother and if mother says 'no', then try to ask separately his/her father and receive 'yes' answer. If you allow this to happen, the child will learn that rules don't matter and it will actively try to break them by asking the right person for permission.
Now if it happens that the father didn't know about what the mother said because e.g. he was in a different room in the house then you need to immediately explain to the child that he/she cannot go to the other parent if the first parent says 'no'.
Therefore it is important to agree with your partner and her parents that you all will be consistent in terms of allowing or disallowing certain behavior and if one of you says 'no' then other care takers cannot override that decision.
Child Respect of an Adults (Talking Back)
What seems like a disrespect is basically a child's need to communicate and learn more.
Compared to the past, today's kids are exposed to and have access to lots of information from the Internet, games and subsequently from their peers. They are becoming more mature in the sense of understanding the world.
As a result, the dynamics between parents and children are dramatically changing. Kids can now better understand the world around them, including what parents are doing and why (to some degree). However, with more knowledge also comes more responsiblity and parents need to let kids to be more independent, more curious and have its own opinions and reasons within certain boundaries appropriate for kids age and surroundings.
However we always need to ensure that kids are safe both physically and mentally from interating with the Internet and continue to have basic rules of behavior.
Therefore we, parents, need to adjust to this information age and understand that having more communication between us and our kids ('talking back') is a good thing as it lowers the chance of misunderstanding and allow kids to better cope with the information.
Child Asking "What are you gonna do?"
Consider the question "What are you gonna do?" as a part of regular communication between you and your partner's son. Calmly explain him why do you consider his behavior to be incorrect. Then you can outline what the consequences will be.
This assumes that you already had a good conversation with your partner and her parents and have an agreement on being consistent and agree on what consequences are appropriate.
So the first step would be to find a time when you and your partner (her parents) can privately discuss this subject and further clarify your concerns