The infidelity needs monitoring, but the drinking needs it even more right now in my opinion. He is driving children around in his car with him.
Maybe it will do you some good if he lost his license... it will keep him put for a while mabye...
Both the drinking and the cheating are problematic, but the drinking you have all the INTEL you want there...
These are times when you have to make some hard choices. I hate these kinds of choices. Any good AA programs in your area? He likley wouldn't go to a permanent place to dry out even if there was an intervention. Often a LOT of DAMAGE has to happen before they are willing to do that.
You could divide up the days when you help the kids out and he does... MTW its him and Thu Fri Sa its you or something like that. The kids may start to see a pattern where he is never reliable but you are. You can't keep cleaning up his messes. I supsect you cleaned up after the dogs... that's classic enablement right there.
You need to do some reading on addiction enablement to protect him and you from that. YOU are at high risk of enablement right now.
Is there any chance you can get his parents to refuse him drink at their place? I dunno, someone should say or do something if he's driving children around to events in his car.
If he's at work drunk he may get reprimanded for that too.
He's certainly dealing with a lot emotionally, but so is everyone his age... I don't sympathize at all... its selfish indulgence at best. But it is very dangerous.
