The Trim tool can add blank space to the end of a Region, in case you want to then pencil in notes..

So how can I pencil in notes by trim tool? I put the notes on piano roll. If I don't like the notes, I just delete them on piano roll and put new ones instead of old ones, isn't? So the unwanted region automatically disappears or changes to new one. Right? Why I need to stretch out then the region with a trim tool.. ?

You don't enter notes with the Trim tool, you do it in the Piano Roll. Remember, the Trim tool isn't specifically for adding blank space, it's for adjusting the length of a Region in either direction. So, for example, if you have, say, a Region that you want to Loop with the Loop Pointer and it's not the right length to loop and stay in time, you can adjust its length [shorten or lengthen it to make it an exact number of bars] with the Trim tool.

I still don't understand. Pencil in means re-arrange notes, right? You say if loop isn't the right length I can use trim tool. What it means? I can stretch the loop as long as I want, so the trim tool. It's the same then. So where is the sense to use trim tool instead of loop pointer? A trim has no notes.. ? Why I need an adjusted length of a region.. ? I already wrote about changing notes (and thus changing regions) on a piano roll.. I don't understand, why I have to use a trim tool. Loop is OK - a repeating of the notes..

You don't have to use the Trim tool. If you don't have any reason to extend or shorten the length of a Region with the Trim tool, then don't use it -- if it ain't broke don't fix it.

How can I duplicate the regions? Is it the same as copy and paste? I know I have to select the region and then hit cmd d, but it doesn't work. What I do wrongly?

You can duplicate a Region with select/Copy/Paste or by Option-dragging.

The video about multi takes. I understand how to make takes, but how to compare them? I mean if he splits and changes a number, then the recorded sound (take) remains there. If I do so, it disappears on the first half of region. The first region is getting empty or most of the take is disappearing from there. So he could compare takes, I don't. Why? Video is included. https://youtu.be/jMLklP7vWd0

In your video you appear to be doing exactly what he did in the tutorial, and it seems to be working properly -- I don't see any disappearing takes. If you're referring to what appears to be an empty Region when you switch the middle Region to Take 1, then there are probably no notes in that section of Take 1 [a note that starts before your cut won't show up in the cut Region -- the start of the note has to be within the Region boundary to play/show up].

Well I remembered that there are notes in the first take 1, but decided to check it out. I highlighted both take ones and re-united them to one complete take 1. (cmd j). Now the notes of the second take 1 also disappeared to nowhere. I have 1 empty take 1, but I recorded it and previous video of the second part of take 1 also shows it. I can include a result. Before the two half take 1s re-united, a warning appeared on the display, but I don't remember, what there was. https://youtu.be/XOhrtiwVFEc