This post contains one affiliate link.
Vanessa, I so understand your questions. Let me take a stab:
"Are my children "only" responsible for their own things (picking up, etc.)?"
If we look at this question Biblically, then the answer is, of course, no. We are never responsible only for ourselves because we are to look for ways to serve others. But we can not force a child to have a servant's heart; that change is God's doing. So we say something like, "Hey, bud, let's not just pick up our own stuff. When you see someone else's shoes on the floor, pick them up, ok?" If the child still doesn't pick up after others (mine have a hard enough time picking up after themselves), then we gently remind them, giving them grace because we know how many times the Father gives us grace and how many times He has to remind us to do something, too. I'm 41 and I *still* disregard what God tells me to do.
Rules are good, but have no power to change our hearts. Rules will reform a heart but grace is the only thing that can transform it.* We want to see transformed hearts in our kids but only Jesus can do that. My job, then, is to point them to Jesus. It's to remind them of the Gospel and their need for the Savior. It's to model the humility that comes with my own repentance before God and them. So we make the rules, we stand by the rules, but we don't expect the rules to produce anything but order in our homes. Grace is the only thing that will make a heart change in our lives.
"Is there still a chore chart, but loosely followed? How does your 7-year-old know to empty the dishwasher? How do I communicate with them in a way that isn't law, yet teach them responsibility?"
My kids all have daily chores, and extra chores on Saturdays. They are expected to do them. They are expected to obey us, because the Bible says that God commands them to obey us. Like every rule God implements, this is for their own safety and good. But they fail at these things, and usually daily. Here's where I get to lavish them with grace and point them to the cross! It usually looks something like this:
I warn the child that they haven't done such-and-such. I let them know that I expect it to be done by a certain time. If I am disobeyed, I make sure they follow through and then I typically have a conversation with them about what God expects, what we expect, and how we just can't do these things on our own. I remind them of the Gospel and how Jesus didn't leave us without help. "We can't do any of these things without the help of the Holy Spirit, can we?" Usually the child acknowledges this, and we pray together. Sometimes I require a consequence, too, such as an extra job that would bless the family somehow.
But sometimes I yell. Sometimes I stomp my feet. Sometimes my kids scatter to the corners of the house so they don't have to hear me yell. Then I have to repent and apologize and ask for forgiveness. Then I have to acknowledge for myself that only grace can change my heart, too. I have to remind myself that while I was still a sinner, Jesus died for me.
These conversations of grace take place all day long. They start at the breakfast table. They come out when someone is struggling with their self image and we have to remind them in Whom their identity lies. They trickle over our schoolwork and when we worship together on Sundays and when we get together for our Community Group mid-week. I want my kids to hear that their faith is about Jesus and what He did, not about what they must now do, because guess what? They now can't do anything outside of the power of the Holy Spirit. I want them to understand that they were saved by grace and that they are maintained by grace.
Because the conversations of grace and Jesus are a daily thing, the conversations about the Gospel which follow a time of discipline are not a new thing to a child here. When I pull the 8-year-old onto my lap and remind her Whose she is, she knows it. She owns it.
*from Give Them Grace, Elyse Fitzpatrick and Jessica Thompson