The bookmarks weren't saved but I was able to salvage them this way, writing instructions if anyone else needs them. (And I'm using Windows 10)

BEFORE uninstalling Opera

(1) If you have bookmarks in your unsorted bookmarks folder, transfer them into a new folder.

(Why? Because Opera Beta does not have an unsorted bookmarks folder. You will not be able to find those bookmarks on Opera Beta. However, when you go to the websites, they will have a heart as if they are saved, but you won't be able to delete the heart so that you can re-add it. It will create a problem where not only are these bookmarks lost, but you can't even re-add them)

(2) Go to this file location:

C:\Users\my username\AppData\Roaming\Opera Software\Opera Stable

The part that says "my username" you will have to fill in with whatever your username is.

Or you might be able to find it here

C:\Users\AppData\Roaming\Opera Software\Opera Stable

(3) There are two files called "Bookmarks" and another called BookmarksExtras. Also a file called Shortcuts and Shortcuts-journal. Copy them into a separate folder on your desktop.

(4) Then you can uninstall Opera

When uninstalling Opera, DON'T tell it to forget your user data, just in case.

AFTER uninstalling Opera

(5) Install the Beta version.

(6) Now go to this file location

C:\Users\my username\AppData\Roaming\Opera Software\Opera Next

If that doesn't work, just go to

C:\Users\my username\AppData\Roaming\Opera Software

and go into the other Opera file (besides Opera stable)

(7) Now go to the folder where you saved copies of the bookmarks.

Copy them into the Opera Next folder.

Make sure that Opera Beta browser is closed when you do this.

(8) Now open Opera Beta browser, and your bookmarks should be there!

Also, can you please post the link for the version of Opera that solved your problems?

Thank you for your help.

[A little tip: whenever possible, try to download programs from their original sites.]
I know! And I usually do so! But there are special situations when I ignore it...CNET is reliable guy I guess!
...

Cnet and its Download.com site have been solidly accused of a variety of adware/malware propagation techniques over recent years, including the deliberate bundling of adware and malware with their download installers. See: http://www.howtogeek.com/207692/yes-every-freeware-download-site-is-serving-crapware-heres-the-proof/ A subsequent study in July 2016 by HowToGeek showed that Download.com no longer was including adware/malware and had discontinued using its own installers. Nevertheless, user trust once broken so callously should only be re-extended very, very cautiously. Frankly, IMO, because the removal of their adware/malware bundling seems to only have been the result of negative publicity, it's way too soon to place much trust in that site yet.

Various studies over the past few years have shown that many other, but not necessarily all, popular free download sites have also been guilty of bundling unwanted software with user-selected downloads. Sometimes the bundling has opt-out choices, often rather obscure, but at other times there is no visible option, and the adware/malware gets installed right along with the desired software. If one stops to consider, there has to be some way for freeware sites to pay their bills, and bundling adware/malware is an all too tempting alternative for some.

It definitely wasn't case if you mean my machine! Though I am getting the point. Thank you very much!

So how to fix it if it is corrupted?

Rename/delete it so a new one will be created.

And - how it got corrupted?

There is no way to know for sure. Maybe a crash, a error while writing some data, there are lots of possibilities.

I know! And I usually do so! But there are special situations when I ignore it...CNET is reliable guy I guess!

[Beta uses a different profile folder so if it's working then I would say that your Opera stable profile folder must be corrupted. Or maybe Opera stable installation.]

So how to fix it if it is corrupted? And - how it got corrupted? Just don't say I have to re-install Windows,please!

since I got a beta version from Cnet site

A little tip: whenever possible, try to download programs from their original sites.

Since I am not so savvy in computer things I still can't understand what went wrong

Beta uses a different profile folder so if it's working then I would say that your Opera stable profile folder must be corrupted. Or maybe Opera stable installation.

Over last two weeks I've been checking manually Opera updates,but there were no any.

Stable versions aren't update often

I can't understand why does searching engine offers old Opera versions?

Well, searching for 'Opera browser' on Google got me Opera's page as first result. And latest version can be downloaded from there.

No! It doesn't!
I did installation from official Opera page,but after installation (and restart) there is no shortcut on the desktop as it is usually.

In a standalone installation there should be no shortcuts.

When I go to /control panel/uninstall or change program list Opera is there,but-there is no way to uninstall it for some reason

Either your installation has a problem or Windows messed up with something.

Have you tried a reinstall? And what about a clean profile?