Oh Man. I dealt with the interrupting thing with skids myself. Constantly, it was "dad...dad...dad...dad...." while we were talking, until my SO finally said what? He is able to tune them out and didn't really notice it, but I finally snapped and said, Just WAIT guys, we are talking, you do not interrupt!. Afterwards I talked to skids myself and said, you guys just need to work on manners- no interrupting when others are talking. Got it? They have been much better, but if they do it, I just hold up my hand without even looking at them and continue talking to my SO until we're done. THEN we look at skids to let them know it's their turn. It's hard to argue with a hand held up in front of your face. If that doesn't work for your skids, then up the ante- if someone keeps interrupting, they get to go sit in their room for 1/2 an hour (or anything that gets it through to them that it's not ok). It sounds like your wife yells at them, but doesn't truly lay down the law and tell them that certain behaviors are unacceptable. If there are no real repurcussions for their actions, or if they're simply told "stop" repeatedly, they are just going to keep doing it.

When skids start arguing, my SO's tone leaves no room for them thinking they can continue. If he tells them to knock it off,  you know he means business. Therefore, he doesn't have to say it that much. They know not to even start.

Some of these things can be addressed by setting rules that leave no room for argument- Everyone sits at the same spot at the dinner table, every single night, and we don't want to hear any arguing about it. We each have our own set spot at our table (it just kind of happened), and there is nothing to argue about. Oldest showers first, every night. Nothing to argue about.  

I wouldn't have patience for all of that- if my SO didn't handle it, I would step in and handle it myself.

(FYI - FIANCEE refers to a woman who is engaged, FIANCE refers to a man. Sorry to be Ms. Grammar Person)

Please don't marry her without a lot of conversations about parenting style and how misbehavior will be handled. She needs to parent her kids.

To be honest, I kind of think my parents had a pretty good approach to my brother and my bickering.

1.  They didn't referee our disagreements.  If you can't get along.. you can't be together.. so we would BOTH get timeouts... or similar consequences.

2.  They were not going to allow an item to be a bone of contention.  Either we learned to share or the item was removed from BOTH of us.

3.  Tattling did not get any response unless the behavior was patently dangerous to one of us (or the home.. like playing with matches).

Unfortunately when mom swoops in and tries to fix and be responsive to every ask.. she has conditioned them to continue to behave this way.

My mom literally told us to stop circling her like sharks..lol.  She also encouraged us to figure out things on our own.  We were also told to not interupt unless it was life or death.. not all the time.. but if my parents had guests over or were doing something.. we knew we were not to pester them.

I think you need to just do it without talking it over with mom. Teach them how to behave with you.   If she doesn’t like it, then she can handle it but for her to allow you to be annoyed isn’t very respectful on her part. 

Ask yourself....IS this how I want to life my life between 'now' and 'dead'?

Tell her to give her x full custody of her kids then you will get married. 

Your SO is going to have to extricate her  head from her own ass and force this kid to grow up.  If she doesn't this is not a marriage I would want to be part of if I were you.

So, figure out the Rx for her Cranio-Rectitis and give her strong regular doses of it until your SO finds clarity.

You can confront the SD-9 behaviors as well. If SO doesn't like how you parent and discipline, then she can step up and get it done before you have to.

Good luck.