I am new to Dart, installed the language SDK and Dartium, followed instructions to run Dart code inside it and it works.
Is that a recommended way to build and debug my client webapp using Dartium and then compile it with dart2js at only final stage?
Should I catch, debug and fix all bugs in Dart source code and then be sure that it won't fail when JS code is compiled and run if it runs good in Dart or is there still a stage of debugging it as JavaScript after Dart part is done?
Usually during development and testing Dartium is used, then before deployment, test your app in different target browsers.
There are occasionally issues that only occur when you run your applications built JS in the target browsers.
It's also worth tuning the build output size (for example introduce lazy loading, ...).
Often also CSS needs to tailored to look the same on all browsers. This is nothing Dart can help you with.
Currently DDC (Dart Development Compiler) is work in progress that allows fast change-reload cycles for Chrome or other browsers that support ES5 or ES6.
You can also use Travis or other CI systems with or without Selenium to frequently run tests on all kinds of browsers.