What do you mean “capture area”, it can be done with chimney going up or vented out (as I was planning to).

For the other one… I think I saw somewhere the instructions to build the frame attach it to the wall (probably plywood in the back too) than install the stone (or porcelain in my case ) over the frame. Am I right?

Those both appear to be quite poor options (and as mentioned above, partially because of capture area).

More: https://bamasotan.us/range-exhaust-hood-faq/

"capture area" means the horizontal space from which it draws air. a vertically-oriented fan will capture very little of the cooking effluent or gas burner byproducts.

En brief: The function of a kitchen hood is capture and containment of cooking plume effluent. Capture is where the plume or part of the plume gets into the hood; containment is where the captured part is entrained in the airflow and pulled past the hood baffles (or mesh filters) and thence outside. The capture area is the area of the aperture that allows air flow into the hood. Capture volume/containment volume/reservoir is the volume between baffles and bottom edge of the capture area. Make-up air (MUA) is the return of outside air matching that which was contained and expelled.

In general, hoods do a poor job of pulling in air/effluent much beyond the edges of the hood, depending on various factors. In particular, they have negligible "suckage" (as a correspondent here once coined) down near the pans. Hence the need for an enveloping shape that the plumes rise into.

As it happens, hot cooking (vaporization point of grease and oil temperatures) creates plumes with relatively high velocity. 1.2 m/s has been recorded in testing. To assure that such volumes of air and effluent do not reflect out of the hood (due to conservation of momentum) instead of being contained, combinations of reservoir volume and intake air velocity are used.

The most efficient hoods are commercial hoods, and while not often practical in residential kitchens, hood designs that deviate too far from commercial design principles will do a poor job of capture and containment. Hence, the earlier upside-down cup comment by @User. There are a bazillion [technical term] hoods designed for style that do not perform well in actually capturing and containing.

Residential hoods having inverted cup like features approximating commercial function, but having more constraints than commercial hoods (except efficiency, which is reversed), need air flows in the region of 90 ft/min entering the hood to assure containment of what is captured.

90 ft/min times the capture area (sq. ft.) determines CFM required. Higher blower rated values than the required CFM are needed for blowers because they can only move their rated CFM in free air (zero static pressure).