Choice of mosfet transistor for linear operation (voltage controlled dummy load / current source)
Why this is documented like this? I should not use this transistor in
this area (low Vgs)? It is not stable at this region?

Yes, it's not thermally stable in this region. For instance, in the IRFP260, if you applied 4 volts gate-source voltage at 25°C the initial typical current will be 1 amp but, if the voltage across the MOSFET was more than a volt or so, the power dissipation would cause the MOSFET to warm. As it warms it draws more current and that causes a massive uplift in warming and, about a millisecond later the junction is at 175°C and the MOSFET is taking 10 amps and dissipating one hundred times the power. It doesn't end well either.

That is what that graph is telling you. Of course you have your MOSFET in a closed-loop control but, if that closed-loop control isn't fast enough to cope with the sudden ramp up of current (and, it really can take less than 1 ms for this to happen), then the MOSFET will be wasted and, it might not even feel warm to the touch. And, you'll be left pondering what happened.

It works well for the purpose for which it was built, but I probably
should add some capacitor at op-amp noninverting input.

General note - when using MOSFETs as a controlled current source you need to run a tight ship in the control loop or they will die on you. Adding capacitors may stabilize your loop electronically but they may add so much time-lag that the MOSFET dies due to thermal runaway (as described above).

Can I use big transistor (like BUZ11A) to regulate small currents?

Sure you can but you have to watch what the leakage is of the BUZ11A because it is specified at up to 1 mA of leakage. Read the data sheet and look for \$I_{DSS}\$.