Are there some activities in research for replacing the vertical stabilizer with a turnable horizontal stabilizer? (flight direction stabilization like the birds)
There have been various (and of varying degrees of success) some flying wing designs that have made it into production with no vertical stabilizer. However they had no turnable horizontal stabilizer either. The B-2 Spirit comes to mind,
The closest thing I can think of that has no proper vertical stabilizer but still does have an epenage is a V tail design, a variety of planes have been made with them over the years including the popular early Beech Bonanza models. (source)
I am not sure of any planes that have ever been produced with such a tail nor any research projects on such a design but that does not preclude their existence.
A general note on aircraft research and design, the various flying wings and V tail planes built over the years have had issues and been met with hostility by some critics. This is not to imply that they are poor designs however the flying wing does have its share of instability inherent to the design. The V tail gained some negative reputation on the early Bo's because many of them crashed. However many people account this to low time pilots in overpowered (for the time) planes.
What you describe implies the use of artificially enhanced stability. There have been projects which did away with a vertical surface completely, lately to improve the low-observable characteristics. However, in each case you replace something that automatically produces a stabilizing effect with something that needs to be positioned actively in order to have a stabilizing effect. So far, the preference for safety has meant that active stability augmentation has not made inroads into regular aircraft - the added failure mode is simply not worth the gain in aerodynamic qualities.
An early example would be the Horten II motor glider (picture source)
A more recent example is the unmanned X-36 (picture source)
In both cases the designers simply left the vertical tail off and used other means for directional stability. Your proposed moveable tail surface has never been tried, as far as I know.