Multiple voltages and currents required in one circuit
microcontroller - 7-12V, v.low current


This probably already has a linear regulator on-board. A linear pre-regulator may be OK at very low current, eg. a uA7812 would dissipate 350mW at 50mA. 


  2x Servo - 6V, 250mA


I bet they draw more than 250mA when moving. You probably need at least 2A peak, so a switching regulator will be required. The LM2576 does 3A and only needs 6 external components (1 diode, 1 inductor, 2 resistors, 2 filter capacitors).


  Solenoid - 18V, 14 Ohm coil


This can be run direct from the 19V supply, with a ballast resistor to drop the excess voltage. A 1 Ohm 5W resistor would reduce solenoid voltage to just below 18V.    


  Geared motor - 9V, 6A stall current


This will require a switching regulator rated at 6A or higher. A switching regulator lowers input current in proportion the increased voltage, but even so it will still be over the limit of your power supply. You need 9V*6A = 54W / 85% (estimated) regulator efficiency = 64W, but the power supply can only put out 19V*3A = 57W. 

You should either get a bigger power supply, or use a separate 9V 6A+ supply for the motor.