As someone who has experience with this, I'm going to chime in.
As has been said before, you need to prepare like a mother fucker FIRST. It seems douchey, but first and foremost secure your kids passports and store them somewhere (out of your house). This means their Japanese passports especially if they have them. A lot of people here can attest that by playing the numbers, her culture has taught her what's acceptable behavior in this situation is vastly different to what yours is. Racist? Probably, but I'm basing this off of experience. To put a travel ban on them my guess is you're going to have to at least file for separation or make a police report. I'm pulling that out of my ass, but I don't know how else the authorities would prevent it from being abused. Of course in Japan the burden of proof of innocence is on the gaijin. Sorry, you have rules to play by.
Do not call her boss. He may tell you what you want to hear at best. That's about it. The absolute best thing you will gain is some time with him not trying to get into her pants so you can work on things, if you want to do that. However, she wasn't attempting an affair b/c of anything he did... she is dissatisfied at home. Sorry for the tough love, but she said it as plain as day apparently. If you want to fix that, it might work, however it will be an uphill battle with cement blocks on your feet. You've also shown you don't trust her (rightful suspicion or not) and you probably never will again. I know I wouldn't. If my girlfriend's husband called me to give me a piece of his mind I would point out the fact that nobody has an affair if they're happy at home. Usually they're quite miserable. Everyone has their breaking point. Maybe you guys just grew apart, but if there's anything in the past that she considers abuse (this includes a lot of stuff... not just physical violence) then it's likely over in her head. I was "done" with my wife for months (maybe years) before I finally left. It's amazing the torture we will go through to be near our children.
Not sure what the working situation is, but you need to document the family assets. Joint and non-joint accounts need to be taken into account. If she has assets in Japan, dream on, but you should still try. She works though, so that's a good thing.
In my non-expert but experienced opinion, things are not going to work out. She's not satisfied, and you're not going to trust her. At least statistically. If you guys can work it out, party on. Having said that, it seems the only reason you guys are together is for the kids. You live in America though, and there is very little stigma surrounding divorce these days. Fuck the telephone bullshit. Talk to her in person, somewhere, and figure out what's best for your kids. They really are all that matters. She says she doesn't want to break up the family. If she means that, she'll be honest and work with you. Be prepared that that might mean separation (probably) or divorce (also likely), but that she'll want to live close, share custody, etc. Trust me - that is a fucking paradise compared to what can happen.
The other huge thing in your favor is that Japan finally signed the Hague convention. So Japan will (supposedly) honor the terms of your divorce should you go that route. That means you need to get a geographic restriction on where she can live assuming she gets custody of the kids.