Custom HTML field multilanguage description

Hello.
I've created a Custom HTML field. Under the description tab I wrote some JavaScript and HTML code that works fine.
But I've also added a language string, i.e. in the HTML code.
Then I've created language overrides for this custom language string.
But it's not translated and remains as on the Registration page.
What did I do wrong?

WYSIWYG editor is likely adding paragraph elements around your string. Switch your editor to None or use the View Source button of your editor to confirm and remove the surrounding elements. Language translations do not work inline.

I'm not using any editor.
But as you said

Language translations do not work inline.

I had it like that:
and I've change it to this:
Still nothing changed. Either I have to wait for cache clear, either there's still something wrong.

Neither will work as I've explained above inline translations don't exist (in Joomla or CB). You need to translate your entire description. Example as follows.

Description: MY_FIELD_DESCRIPTION
Language Key: MY_FIELD_DESCRIPTION
Language Text:

Got you.
I've also saw that putting "required" option ON does not work for HTML code field and there aren't ways to get icons at the right of this field on registration form.... Is this normal?
However I've added
class="required" in my code under Description tab.

I've no idea what you're trying to do. Are you trying to add an input directly in a Custom HTML field? There will be no storage behavior for that field so that isn't going to work. As for the required class it's checked by our validation if on an input. Icons in a field are just inline elements (spans) next to the input.