I mentioned the minnock case because it is an extreme version of parents not acting in the best interests of the child.
Ms minnock was determined not to allow her son to see his father and the press became involved when she ran away because she would not comply with court orders regarding access.
here's a link with PART of the story.
https://www.theguardian.com/uk-news/2015/jun/24/rebecca-minnock-voices-fears-of-emotional-impact-on-son-if-she-is-jailed
remember this is an extreme case but it chows the courts are no longer tolerating either parent who doesn't put the child's interest first.
If you accept that the court wants children to have access to both parents and that is the child's right then maybe you can approach your mediation session with a different frame of mind.
Put your son first but if you strongly believe that he is in real danger from his father then gather evidence as best you can.
The dog issue can be resolved because the court would just make an order that he is not allowed in the same location as your son - every other issue may have a solution & this might include parenting education and anger management for your partner.
The contact with his father wont be prevented unless there is a real danger to your son.