Hi Mate , no criticism is intended 're drop the rope. I personally took years to really let go and even then I sometimes feel like it's there a bit.
Your sitch is tough , W comes to your home , you use her car , you get dragged into her life 're hospital etc. That's not making it any easier for you.
Just a suggestion but how about not letting her into your flat but she collects kids from the door or you bring my them to the car ? She's created an atmosphere now with the constant threats and in doing so given you a reason to calmly suggest that to avoid further upset or arguing , the child exchange be done outside of your flat.
're the mediation, how about yhou arrange it and then give her the time and date. If she refuses then you may have to go the expensive legal route but you will have shown you tried and that could stand to you in the future.
Just my thoughts mate. Your kids are just that , kids and they need to see you as calm as possible regardless of W and her moods. I know your a great dad and you've kept your cool where many of us wouldn't have so your to be admired for that.