1. I have encountered small vertical jitter. How can I solve it? (Of cource, I can filter this value, but I'm looking for more accurate method).
Jitter can arise from a variety of sources. Here's an incomplete sampling:
-It might be that the vertical force continually pushes the character beyond the reach of the force, allowing it to fall back down repeatedly.
-It might be excessive spring rigidity combined with a long time step.
-it might be numerical issues with the convex cast and the environment.
-It might be some order of operations issue.
I can't really say for sure without spending more time analyzing it and seeing it in action (which I would prefer to avoid

).
2. Can I implement this method more sophistically using some constrain or jointed body or something else that provided by BEPU physics library?
It can be implemented more robustly in the form of a constraint like the VerticalMotionConstraint in the CharacterController. Doing so requires greater familiarity with the solver. It may not be very useful to spend the extra time doing this unless you plan to have lots of physical structures dependent on the the character. There is no premade 'hover constraint' in in the main library.
3. Whether this method can have pitfalls that I have not met?
There's a fundamental tradeoff with stepping methods based on forces and velocity:
-If it moves too quickly, the velocities will be high enough such that it can cause explosions. Stepping near some object could send that object flying. In certain cases where the character loses contact with the ground, the character might go flying.
-If it moves too slowly, the character won't climb fast enough. The body will bump into the stairs, halting the character repeatedly.
That tradeoff can be manageable depending on the environment design.
However, there's a more dangerous combination of problems:
-Velocity-based stepping does not avoid the need for step validation in the general case. The character will still happily step into invalid states unless explicitly prevented.
-"Hovercharacter" methods, where the character has open air below it for stepping, make it extremely difficult or impossible to stop invalid steps. Since there's nothing stopping the character from moving horizontally onto the invalid support, they will happily scoot into an invalid state. At that point, there's not really any good way to recover.
While it's possible to address the second problem by removing the hover margin, other problems are introduced. Without the hover margin to smooth out steps, the character will just slam up against the stairs repeatedly while waiting for the force to push it up and over. A more complex approach built around conditionally active collision constraints could address this too, but there will be a swarm of corner cases to handle.
I need little bit different dynamic "feel" of character controller.
Before going through the hurricane of corner cases associated with full-featured character controller development, I would strongly recommend seeing if the existing robust CharacterController could be modified to do what you need. Keep in mind, a lot of the 'feel' of a character controller can be defined by factors unrelated to the actual physical representation of the character. For example, smoother stepping can be accomplished by loosening the camera interpolation.