I'm answering here the questions that you cannot google answers for yourself:
This is because sending precompressed version uses less server resources then gziping on the fly?
Yes. Don't worry about the use of mod_deflate elsewhere in your site, it's really not relevant.
what about other logged-in users, commenters, pages excluded from caching and every other situation where minification is done on every page load?
You're not understanding the point of the plugin. You can disable HTML minify if you want and in fact is not enabled by default. Disabling it for the cases you suggest defeats the purpose of the feature. Disabling it for administrators was to simplify configuration of CSS/JS minify groups.
Should files be added by copying whole URL (eg. http://example.net/wp-includes/js/jquery/jquery.js?ver=1.3.2) and are they after that removed only while they are with exact URL, or are they deregistered from WP (eg. what will happen when jQuery in WP is upgraded to 1.4.2: will that new script be minified and removed or only old one because of different URL)?
The FAQ does explain this. You should put the entire URI for the file in the input box when a GET string is set for it. If your HTML source outputs
http://example.net/wp-includes/js/jquery/jquery.js?ver=1.3.2 then you can use
wp-includes/js/jquery/jquery.js in your settings without issue. Regardless, use the test URI button to make sure that minify can find it. If you update some files in your templates, you need to update your settings. Minify is obviously matching the files that exist in your theme's output.
What do groups mean? Are those files we add minified only for that group or what?
The minify settings page clearly shows groups label followed by a drop down selected list of templates. The default group means that the files specified appear in all templates and don't need to be specified for individual templates; the other templates may have their own CSS/JS for which you can handle using the file group for template.
What does non-blocking means for JS?
It means that scripts will be embedded in such a way as to try to make them load asynchronously without disrupting the page render.
What happens when already minified JS file is added: will it be tried to be minified again or script will recognise minification?
Try and find out. Some files cannot be minified at all for numerous reasons. Whether or not a file is minified more than once is really irrelevant. Some can only be combined. You will have to experiment to determine what's possible + optimal with your theme/plugin combinations.
If we turn on cache control headers, where to setup max age of cache?
It's done for you based on the expiration time specified in your settings.
What is difference between "Maximum lifetime of cache objects" and "Garbage collection interval"? When someone looks at it, he can conclude that both things are related to how often W3TC regenerates caches of pages.
Just as for most of your other questions the captions explain. The lifetime of the object means how long an unchanged cache file should be saved and used. While garbage collection determines how often expired files are removed. There is no other way to explain this.
To what external files "Update external files every" applies?
Again I would have hoped the caption is clear. External files refer to those you do not host that you've specified in your CSS/JS groups. We cannot assume that they will never be updated, so minify will obtain those for you at the specified interval and use the update file the next time the cache is built.
When pages that shouldn't be cached are added, should we add them like http://example.net/page, /page or simply page?
/page/ is fine, regular expressions are also supported.
I read that cache is regenerated when new entry or edit was made but what if something else is updated, like RSS widget, Twitter etc?
It depends on how those are implemented by the plugins you are using. You can use fragment caching in an upcoming release or use a low expiry time for your page cache files to frequently generate new cache files.