No, a darlington doesn't give you more "power handling" than a single BJT (bipolar junction transistor, these are the ones that come in NPN and PNP types). In fact, a darlington is bad for power handling due to its large voltage drop when on. This causes much more dissipation at the same current as a single BJT.
The only advantage of a darlington is that its current gain is much higher than a single BJT. It is effectively the gain of the two BJTs making up the darlington multiplied together. This can be useful when switching low currents controlled by high impedance signals, and you don't need high speed.
There are other ways to start with a high impedance signal and provide enough current to drive a single BJT switching element.
As for the distinction between MOSFETs and BJTs, each have their advantages and disadvantages. BJT are controlled with current at a low voltage. Any BJT can be driven with logic-level voltages. FETs are voltage controlled, and all but some relatively low voltage FETs (up to 30 V or so), need 10-12 V gate drive. That requires a special FET driver chip or circuit to control the FET from a typical logic level signal.
Both BJTs and FETs can handle significant power in the right cases. BJTs look more like a voltage source when on, and FETs more like a resistor. Which one dissipates less power depends on the current and the Rdson of the FET. At a few amps and 10s of volts, FETs are more efficient since the current times the Rdson is less than the 200 mV or so of even a well-satured BJT. The FET voltage drop goes up linearly with current. The voltage drop of a BJT starts out higher but goes up less than linearly with current. At high currents a BJT can drop less voltage. Also, FETs that have to withstand higher voltages have higher Rdson, so BJTs look like a better deal at higher currents and voltages. When dissipation and a few 100 mV drop isn't a big issue, it comes down to price, which BJTs are usually better at for equivalent voltage and current capability.