1. How often should Wordpress be updated?
As often as there are security updates. So, yes, when there's an update for WordPress, you should take a look at the changelog if you are not planning to go with the update.
2. What is the best way to do it?
A. Backup your database. B. Then, if it's a security update, you just NEED to go with the update process, no matter what. C. If it's NOT a security update, I'd check the compatibility of the plugins I use with the latest version of WP. For instance:
3. How to know if there is a problem with plugins? When a new version of Wordpress comes out, we have no way of knowing if there is a problem with existing plugins.
You can never be sure. I've read that, in it's history, WordPress has never removed depreciated functions altogether (but that does not mean, you are all set), so you are probably safe until the plugin author fixes it. (Point (2) also applies here.)
4. When should plugins be updated? At the same time as Wordpress?
Same as point (1). If it's a security fix, just update. If not, update if you need the new features or fixes (again, look at the changelog for the plugin).
5. Is there a test suite that is run before releasing Wordpress? Does it include plugins?
All stable releases of WordPress are very well tested before release. But are not tested against plugins. It's for the plugins to keep up with WordPress, not vice-versa :)