Dipping a Dyson Ring below the event horizon The basic assumption about inescapability beyond the Event Horizon is that the necessary escape speed (orbital speed) would exceed speed of light, therefore no object can achieve it. Now, would it be possible to escape it by other means, say, by mechanical push against the gravity, through application of force without increase of speed?

First, short introduction: What is a Dyson Ring? (feel free to skip or skim this section if you know it):
Dyson Ring is a hypothetical, artificial, ring-shaped structure surrounding a massive body (e.g. a star), rotating - essentially orbiting it - rotating at speed slightly higher or lower than orbital speed necessary to maintain orbital equilibrium - state of freefall near its surface. This way, its rotation creates centripetal force simulating gravity for inhabitants. It would create a habitat much larger than any planet, given some "rim" it could hold atmosphere, and essentially is a neat sci-fi alternative for a planet - similar conditions, vastly more surface.
The one property of it is that tensile strength of its construction allows it to exist at distance from the central body (star) that is off from normal orbit matter moving at this speed would take.
Most theoretized Dyson Rings move slightly faster than needed, stretched out, with the inner side inhabited, and the Sun in zenith for all of their surface. It is possible though (if not recommended due to buckling risks) to make one that moves slower, the star's gravity not fully overcome by rotation, compressive force applied to the construction.

Now - technological problems aside - of building a Dyson Ring (not habitable and much smaller; a scientific device of ~30km radius.) and bringing it close enough to a Black Hole.
Let's imagine a Dyson Ring that can change its circumference (say, built of segments connected with expanding/contracting actuators) - and, as effect, radius within certain range - specifically between outside and inside of Schwarzschild radius of a selected Black Hole. It is also supplied with power sources that allow to vastly increase its rotary speed.
Now, with the actuators expanded the ring is placed around the black hole, outside the even horizon, rotating so fast that its segment orbit the black hole. Its angular velocity so high the linear velocity of its surface is near to speed of light. Normally the centrifugal force would tear it apart, but it remains in equilibrium with centripetal force of gravity of the black hole.
Now, actuators contract. The angular velocity rises a little, with accordance of law of conservation of momentum, but since we're at relativistic speeds, the increase of angular speed is minimal; it's mostly the ring gaining some mass.
Now, the ring is below the Event Horizon. Since at this altitude orbital speed would exceed the speed of light, any object there would quickly spiral into the black hole. Still, the ring does not, due to its own tensile strength preventing further decrease of its own radius. It moves slower than the orbital speed, but the mechanical strain overcomes the gravity. (note, while below the Event Horizon, we're still pretty far from the singularity, so the forces exerted are not yet extreme enough to cause collapse of matter, and the rapid rotation overcomes great most of the compressing gravitational force)
Expand the actuators, increase the radius of the ring - and it is forced back outside the Event Horizon, able to broadcast its findings.
Possible physically, or am I missing something?
 A: This is one of the common fallacies when it comes to both special and general relativity.
A lot of people, when encountering SR for the first time, think that causality can be violated by using a long pole to send messages.
Similarly, it is a common thought that one can "dip" a pole into a black hole and then pull it out. After all, solid things are solid, right?
Nope. The resolution to both these paradoxes lies in our assumption that truly "rigid" bodies exist. Poles seem to be pretty rigid in our everyday experience. But they aren't completely rigid. When you push a pole, the other end does not instantaneously shift. Instead, the local region of the pole shifts, creating a small region of compression. This region of compression propagates at subluminal speeds, preserving causality.

When we dip such a rod into a black hole, the atoms of the material cease to exert forces on each other. Electromagnetic forces require the exchange of carrier particles. In this environment, while the outer metal may send carrier particles to the inner one (pushing it even more inwards), the reverse isn't possible anymore. So the metal loses all rigidity, becoming more like a gas (though at this point atoms don't exactly exist either). If someone on the outside tries to pull the rod out, they'll only get the half that was above the event horizon. The half that was below will have broken off and fallen.
In your case, the expanding ring will have to exert outward forces on itself to expand. However, the inner metal does not have the ability to "push" on the outer metal. So it cannot expand (and this is before we even take into account the disintegration)
A: Your question is very nearly, but not quite, a duplicate of Fighting a black hole: Could a strong spherical shell inside an event horizon resist falling in to the singularity?, and the answer is the same.
The forces that hold matter together propagate at the speed of light. Once at or inside the event horizon the forces cannot propagate outwards, so they cannot resist the inwards fall of the matter. Not even if it's a spherical shell (or in your case a ring).
For a calculation to show that even light cannot resist the inwards fall within the event horizon see Why is a black hole black?.
