I’ll try to insert a picture here that I do not see:

 And one more picture:

Are you seeing this on images from other sources? I.e., not vdraken.clan.su ?

Yes, they are shown in bbpress here you can see it. 

Not what I mean and not really relevant

I was meaning if you insert an image into an SP post from a different source domain do you see that problem.

In general, all the pictures are displayed normally, but nevertheless there are some pictures from other resources that just do not appear.

There may be a code problem in some of the V5 changes to this but I will need to do some tests before I can be sure so that is a task for later today.

But I can explain why you are seeing what you are seeing (even if it ends up that the code is wrong)!

If you are using the popup image enlargement feature to display images, then SP needs to know the width of the image. A normal and correctly formed html image tag will include the width. If, however, the user making the post has not included the image width then we need to interrogate the image file to determine the width. So a call is made to the source image url using the php function getimagesize().

If this call fails then you end up with the message you have seen. Which probably means the server in question was unwilling to provide the data and blocked the access. This is pretty unusual which is why I asked about images from other sources.

Looking at the code however does suggest that we may be making that attempt whether the image popup function is being used or not which is most likely incorrect as if it is not being used and while knowing the image width is useful, it is not an imperative. The one good reason for doing this whether the option is being used or not is, of course, in case a user decides to turn it on at a later date as it will ensure that all images DO have the width recorded in the post data.

I suppose one good way around that problem is to continue to seek the width as we do now but not to fail it if (a) the server rejects the attempt AND (b) the popup feature is turned off.

So I will take a longer look at the code today and run some tests and make those changes if necessary and possible.

Thank you for the very detailed explanation of the problem, so I will use option (b) turn off popup feature.
Is there any possibility to get the size of an image, for example, temporarily downloading it and delete after getting a size ? Are there any other ways to do this?

Downloading would be somewhat expensive.
I see you were using the bbCode editor. If you used the tinyMCE rich text editor and used the image insert dialog (and for uploads our uploader SP Plugin) then this would try and collect that data.
Or, of course, add the image width into the tag itself.

The code for V5 was updated with that change by the way.