How to call function on child component on parent events

Give the child component a ref and use $refs to call a method on the child component directly.

html:

<div id="app">
  <child-component ref="childComponent"></child-component>
  <button @click="click">Click</button>  
</div>

javascript:

var ChildComponent = {
  template: '<div>{{value}}</div>',
  data: function () {
    return {
      value: 0
    };
  },
  methods: {
    setValue: function(value) {
        this.value = value;
    }
  }
}

new Vue({
  el: '#app',
  components: {
    'child-component': ChildComponent
  },
  methods: {
    click: function() {
        this.$refs.childComponent.setValue(2.0);
    }
  }
})

For more info, see Vue documentation on refs.

What you are describing is a change of state in the parent. You pass that to the child via a prop. As you suggested, you would watch that prop. When the child takes action, it notifies the parent via an emit, and the parent might then change the state again.

var Child = {
  template: '<div>{{counter}}</div>',
  props: ['canI'],
  data: function () {
    return {
      counter: 0
    };
  },
  watch: {
    canI: function () {
      if (this.canI) {
        ++this.counter;
        this.$emit('increment');
      }
    }
  }
}
new Vue({
  el: '#app',
  components: {
    'my-component': Child
  },
  data: {
    childState: false
  },
  methods: {
    permitChild: function () {
      this.childState = true;
    },
    lockChild: function () {
      this.childState = false;
    }
  }
})

<script src="//cdnjs.cloudflare.com/ajax/libs/vue/2.2.1/vue.js"></script>
<div id="app">
<my-component :can-I="childState" v-on:increment="lockChild"></my-component>
<button @click="permitChild">Go</button>
</div>

If you truly want to pass events to a child, you can do that by creating a bus (which is just a Vue instance) and passing it to the child as a prop.