How often should Wordpress be updated?
WP should always be kept up to date. You should update it as soon as possible after an update is released.
What is the best way to do it?
Always have a current backup - and know how to restore it if you have to (or have someone handy who can).
Do you make an income from the site? If so, you can set up an exact duplicate of your site (as previously mentioned) and test the updates there first. If you're making an income from the site, it's worthwhile to hire a developer to maintain your site, and go through all these hoops for you. Or, you can try using a WP host that does upgrades and automatic backups for you (as long as you're aware that hosting companies normally don't actually look at your site to make sure it's not broken).
If you're not making an income from the site, you can make a list of all the plugins you use (and what they're for), and all the functionality that your site needs to have to not 'break'. You should use as few plugins as possible, and delete any that you don't need.
You can either click on the Update button, or set up WP 3.8+ to update itself automatically (with options of major updates only, or including minor updates and/or plugins). If you use any automated system, make sure you periodically check your site hasn't lost any functionality (see below). It also helps if you have widget in your sidebar or footer that asks your visitors to report any 'bugs' they find to you.
How to know if there is a problem with plugins? When should plugins be updated? At the same time as Wordpress?
After making sure you have a backup), update WordPress, then update your plugins. Go through your list and check that all your plugins are still working as expected. Is your contact form still working, for instance? Are your shortcodes still working? Are you getting any error messages? If something looks wrong or broken, go to the plugin developer's page and check the support forum. If nobody else seems to have the same issue, do some troubleshooting on your site. Does the plugin work if all other plugins are deactivated? What if you switch themes? Submit a support ticket if necessary. Be prepared to find another plugin to replace the one that 'broke'.
Is there a test suite that is run before releasing Worddpress? Does it include plugins?
I'm not sure what you mean, other than setting up your own 'clone' of your site for you to do a 'practice run' of an update on.