Can you create a checkbox without using a input type=checkbox html element?

You certainly can. The advantage is that it gives you way more flexibility. For example, here's a basic React component that uses divs for the checkbox:

https://jsbin.com/xarewiweza/1/edit?html,js,output

const styles = {
  outer: {
    borderRadius: 5,
    border: '2px solid gray',
    width: 30,
    height: 30,
    cursor: 'pointer',
  },
  inner: checked => ({
    borderRadius: '50%',
    height: 28,
    width: 28,
    backgroundColor: checked ? 'red' : 'transparent',
    margin: 1,
    transition: 'background-color 0.2s ease',
  })
}

class Checkbox extends React.Component {
  constructor(props) {
    super(props)
    this.state = {
      checked: false,
    }
  }

  render() {
    console.log(this.state)
    return (
      <div 
        onClick={() => this.setState({ checked: !this.state.checked, })}
        style={styles.outer}>
        <div style={styles.inner(this.state.checked)} />
      </div>
    )
  }
}

ReactDOM.render(<Checkbox />, document.body)

If you wanted animated checkmarks, or other cool stuff, you'd have to go custom.

The disadvantage is that you don't get some of the default HTML functionality, though I'm not sure if there is anything useful with the checkbox specifically. For example, with the default select on iOS, it automatically uses the carousel selecter, which is a handy feature.