Is it possible to focus the sun in such way?
Yes, as others have pointed out, all of the ideas in your sketch are already used in existing designs - perhaps excepting the shutter (which actually performs no useful purpose so far as I can see).
As Chris White commented - "this exact design (with the shutter permanently open) is a Schmidt-Cassegrain telescope, probably the most popular high-end consumer scope these days."
Is it possible to increase the power of the beam by making it bounce between the mirrors
No, focusing a beam of light, or reflecting it, does not increase the power. The amount of power is the amount of light energy entering the system per second. That is limited by the diameter of the entry pupil. Energy is conserved.