Based mostly on your other responses - you need to be rid of this guy. I don't know if he is abusive / controlling, mentally ill, or what, but he is absolutely no good for you or your kids. He left, don't ask him back. I suspect he is waiting for you to beg him to return. Please don't do it. I don't see anything positive here and a lot that is negative, with a hint of serious danger. Get a lawyer, draw up divorce papers and be done.
He us wrong, over-punishment fail. My parents made that mistake with my sister. She was a troublesome child and they took away all her rights and privileges - and then what? Short of beating her there really was nothing they could do. Punishments have to be measured - and its a really difficult problem. You know your kids and you can do better than anyone else.
It is not reasonable to expect children to only listen to religious music. This is a case where an extreme restriction will really be no restriction at all, because they will not respect the restriction. Lots of kids listen to songs with bad words, sneak peeks at the equivalent of playboy and try a cigarette. Its vital to separate normal stuff from trying heroin, or getting pregnant.
Again, get away from this guy. Don't take him back when he comes back (he will, if you don't beg for him first). He will be all apologetic and loving - its just that you drive him crazy you know......
Mentally Ill, or an abuser? You don't want either.
He would punish them by taking freedoms away. Great! No problem! I know I could do better at my parenting and I could use the help with the kids! However....he would ground them from everything....for anything they did wrong. There is a warning first, but if it every happens again there must be a punishment. Not a single other warning. I'll be very honest....if I punished my kids the way he wants me to for EVERY single thing they did wrong, they would have no possessions and would ALWAYS be grounded. There was no leeway at all. Maybe that is how parents are supposed to be? Maybe this it my delusional issue that he speaks of? So many adults, teachers, coaches, bosses, compliment my kids and how much they like my kids. Can I really be doing that bad of a job and have so many people adore my kids?
Not to mention, with how long I was a single mom and parented my way, it would cause serious rebellion. We tried his way for about 3 months. I was AWFUL. The kids lost respect for him, and they were quickly losing it for me to. It wasn't how our family functioned. My then 15 year old even took off for several hours. I told H that I could not parent how he wanted me to. He backed off for a while. But I think ultimately he wanted me to do it his way. He essentially claims his kids are perfect, even though he will deny it if you ask him directly. But every time my kids would do something wrong he would say his kids would NEVER do something like that....!