The 180 is about changing YOUR behaviors. And not for the marriage, for yourself. Detached from the marriage and the kids? Well, you can't address the marriage, but you can the kids. Emphasize no screen time with the kids (that means no phone for you when you're doing no screen time with the kids. It's hard at the park when you're watching them do the same monkey bars for the fifteenth time, but try!) when you're with them.
For the GAL, don't run out dating, but definitely do new experiences, fun stuff. Try to keep it not extravagantly expensive (unless you guys are made of money) but definitely fun. In your situation, you absolutely cannot go out with other women right now. But do something besides playing video games by yourself. You need adult, human interaction and support, so you need to see your friends often, and not just in a sobbing in their arms (although that can happen, it has for me) and going on about the divorce.
Detach. This is separating the emotions from the current experience. I'm in the same boat, with the love of my life leaving me, and it's hell. But releasing from the emotion of it to look at it objectively is important. Owning and validating your contributions to the breakup is important, especially since you initiated the BD. You are a WAS, not a LBHS, who has decided to try and recon after your wife has moved on, so you're threading a different kind of needle than some here. You need to show remorse and effect honest change in yourself, and hope for the best. Admitting you made a mistake and were an idiot to her was fine, but you can't pressure her too much. It was hard for me to grasp that the marriage is BROKEN right now. It is. I spent awhile trying to find the magic words to go back to what we had, but there's no going back. Same for you. Your marriage is done. If you reconnect, it will be a new relationship, even if you never officially divorce. Your old marriage is gone, and good riddance, because you were a terrible partner (and she made bad choices in it, too.) If you recon, you will build a new relationship and will have the chance to make it far better, open, and caring. So don't eff it up if you do.
You're in crisis mode right now, so follow the advice of the regulars and old timers.
Again, I have to say, do NOT start dating as part of your GAL as YOU are the WaS, and her doing the 180 (initiating divorce) is what knocked you on your butt and got you wanting to recon.
Her 180 probably has nothing to do with wanting to save the marriage, but is about her wanting to improve her situation where she was in a marriage with a man who repeatedly said he wanted to leave.
Do not get angry.
You will get upset. Excuse yourself from the convo as soon as possible. Do not DROP the convo, though, you can come back to it and say you're ready to continue the conversation more calmly. You can say "Let's take 5 minutes" or "let's try again tomorrow" or something depending on how upset you're getting. But stop before you get angry.
Do. Not. Get. Angry.
Validate her feelings. Understand them as best you can. Start therapy by yourself if you can. Let her know you're starting therapy to deal with your issues, but don't phrase it as "So we can get back together."
My recommendations for 180s :
Kids : Involved and invested dad. Support your wife to the kids. Do not ever ask them to spy/etc. Do not blame your wife or trash talk yourself to them.
Therapy : Do it. For yourself. For your future relationship with your kids, and your wife or ex wife, whatever happens. And again, for yourself.
GAL : Drop the screens, do fun things with your male friends/kids. Think about your emotional well being and your kids well being. Exercise, meditation, etc for calmness, doing things with your friends for fun and emotional fulfillment.
Communication : Polite Neighbor for sure. Keep your wife informed of stuff about the kids (updates when you have them) and be punctual and courteous. Avoid all R discussions unless she starts them, then do your best to let HER talk. It will be agonizing to do so. Listen to her. LISTEN. TO. HER. You don't have to agree with things you don't believe to be true, but affirm that you understand that's what she feels. This is a hard line to walk. Walk it. Practice walking it in a mirror.
Me: Mid 40s W : Early 40s M:11 T:13 S, D, both 7-10 BD : 11/2017 Separation : 1/18