So, the question is decided, and not without the presence of magic.
First there was a correspondence in the Microsoft community section of the Linux Skype. Then there I was transferred to the PM and ask clarifying questions.
During the tests revealed the following: send messages in Skype 220.127.116.11+ Linux (all platforms, home and work computers, that is, different network) not working only in my personal account. Other accounts make this easy. All other Skype earlier Linux (5.5), or all kinds under Windows, worked with account fine.
The key point in testing was to connect the computer through a cellular connection to check work Skype when the computer is not through the LAN. And then it turned out that in this connection the Skype 18.104.22.168+ from my account starts normally send messages! At this point the support of the first level States that it is the problem of NAT in the networks that I use, and that technical engineers don’t take it to work. Supposedly this is normal behavior for Skype. I was outraged, of course. They offered to write a review here https://feedback.skype.com/?p=10021&v=22.214.171.124&lang=ru-ru&eTag=8D13EF10C52BC2E supposedly there collected opinions and my opinion will listen.
With grief I decided to find old Skype 4.3 and try it. Found here https://www.techspot.com/downloads/4985-skype-for-linux.html
Installed as the old package:
sudo zypper in --oldpackage ./skype-126.96.36.199-suse.i586.rpm
To solve the problem with entering Cyrillic (wrote gibberish) installed an additional package:
sudo zypper in glibc-locale-32bit
An old Skype works fine. Rename contacts, sending messages, calls and contact group, was in place. Although emoticons are not loaded from the code, remaining the text.
Well, OK, I decided. And was going to install the Skype at work then. I came to work. However, I decided to write a couple of messages in unhealthy 188.8.131.52 Skype. Just to try it.
And then I see that the messages began to go and reach the recipients! Here already half a day, as the same Skype 184.108.40.206 (without restart or re-login) works fine with sending quick messages.
p.s. What there is of 4.3 struck on the server that passed plugging in the new version?