Relationships begun as affairs are some of the most volatile, hostile, selfish and destructive relationships out there. OM is a complete fool to give up a kidney for a wayward woman he doesn't even know as a healthy (un-sick) person. At least you giving her a kidney makes sense, even divorced she's the mother of your children. You have an interest in the child rearing partnership if nothing else.
Part of me wonders...wishes...questions whether this whole thing could somehow be your wife's attempt NOT to risk your life giving her a kidney because her kids need their father forever so she went out and found some sucker to donate one to her...then, maybe she ended up really falling for the guy in the process. I know that's not possible this was a PRIMARY excuse for her - but waywards come up 10-100 different rationalizations and justifications depending on the day to tell themselves it's OK to do what they continue doing.
If OM is at the hospital...and you have the kids and are, I presume, expected to bring them around to visit mom.....I'd consider advising you NOT to take them around under the presumption (which you state later on) that them being around OM is unhealthy for the children since he's the man that interfered in their family and main cause of their parents divorce. It's also hurtful to YOU and YOU MATTER TOO. It's like asking you to come around your rapist.
When the kids are with her she can do what she wants but when they are with you...YOU are under no obligation to expose them to OM like he is some kind of friend of the family (you included) nor some kind of hero (because he foolishly donated a kidney to your wife/their mom). When they are in your custody you also have a duty to protect them from danger. OM is a danger to them. He quite possible targeted a sickly woman with small children because he is a predator. You don't know this guy from Adam. Nobody has given you his name or done a background check on him and the kind of guy willing to pursue a sick married woman is definitely a risk to your children. He lacks character.....so, if they want to bring the kids around him on their watch that's their business (even though they better keep an eye on him and never leave him along with your children). When the kids are with you....they will never be around OM.
Later on...you also may consider a restraining order against OM if he starts trying to hang around your kids at sporting events or school events like he's their step-parent or something (hopefully their relationship never lasts that long but something to keep int he back of your mind). You just make sure your kids never see you accepting any kind of peaceful friendship or even coexistence with OM. That way when you later tell them what a bad and dangerous man he is in terms they eventually will understand, your behavior will be consistent with that information and they will MODEL that behavior the rest of their lives (they you don't have to be nice to abusive people for everyone else's sake).
The internet is 90% complaining and entitlement and I hate it because I deserve better!
