axes and pin configuration for gas cutter

idsoft wrote:

Where I can read about coordinate systems and kinematic, it's desirable with pictures(that explain the naming of the axes and their position)? Or, maybe, how I can use other axes different from XYZ?

The G-code axes are XYZABCUVW.
Normally XYZ are cartesian directions, A B C are rotary axes with A rotating about the X axis, B around the Y axis and C around the Z axis. U, V and W are normally extra linear axes, parallel to X, Y and Z respectively.
However, this need not be the case. You can use any axis as rotary or linear, and there is no requirement that they occupy any particular orientation in space. (though the Axis graphical preview will be wrong if they don't)

If you look in the HAL file for an EMC2 machine configuration you will see that there are "pins" referenced with names like axis.0.joint-pos-cmd. The value of that pin is directly controlled by the "X" word in the G-code. (it's value will be the absolute axis position, not the on-screen value). A "Y" word in the G-code affects axis.1.joint-pos-cmd and so on until W and axis.8.joint-pos-cmd.
(In general the HAL file will use motor-pos-cmd instead, which is the backlash-compensated value)

Probably, this is the perfect solve for me. But, can I re-wire the HAL layer by GUI buttons instead M-codes?

Yes. You can create a gui-button linked to an MDI_COMMAND in the INI file, somethig like
MDI_COMMAND G53 G0 V0 W1 M123
(I am not sure that that would work, as the M123 might be actioned first, but there are ways round that)