1) This script tag is for browsers that support Dart (currently only Dartium) other browsers ignore it.
2) I haven't used it without polymer elments, but if you put your HTML inside a <template>
tag you can use declarative binding like
<input type="button" id="alertBtn" value="Push Me!" on-click="{{showAlert}}"/>
dart-polymer-dart-examples contains several examples how to do this.
You can also use
document.querySelector('#alertButton').onClick.listen(
(e) => window.alert('Hello from Dart!'));