I would like to show first the posts which have certain checkbox, let's call it "First", active. Now if I set it as "Ordering->Order by-> Field - First, the view shows only the posts that have the field activated. Is this intentional or do I have some misconfiguration somewhere? There's no secondary sorting.
When you order by some field, any posts that have no value for that field will not be included in the results.
That's not something specific to Views, it is how WordPress works (because of how MySQL works).
So if your checkbox field saves nothing when unchecked, all such posts will be omitted when ordering by this field.
Update your field settings so that it saves zero when the checkbox is unchecked. That will require you to re-save all such posts for the database to be updated.
Then the ordering should work, let me know otherwise.
Note that there is a known issue with saving zero when unchecked for the checkboxes field type, which you should avoid, but with the checkbox field it should work okay.