Is it possible to connect to EC2 instance directly? This is possible on Rackspace using a java servlet inside your browser where you get a "remote monitor window" (not SSH but a local login screen). I ask this because being able to connect only over SSH can be dangerous. It can happen that something goes wrong with your SSH and you can loose your server for ever. I'm surprised that I don't see this option in my account. What is the right way?
No, this is not possible.
This functionality would be trivial for AWS to implement (several other VPS vendors have done so), but they choose not to, presumably to encourage their customers to use EC2 as it's intended to be used - each instance should be able to be discarded and built from the ground up without affecting your application.