Is it possible to physically measure the input and output impedance of a component?
Is it possible to physically measure them like we measure resistance?


We can use a network analyzer to measure the reflection coefficient of a port of the component, and from this we can calculate the input impedance.


  Is it possible to measure it for BJT based transistor amplifiers and FET based transistor amplifiers? Also, can the same be done for complex ICs like FPGAs and microprocessors and PHY chips?


Yes and yes.


  A signal going into the input of the IC however has to travel on the PCB track, then through the pin on the package of the IC, through the bonding wire and then into pad and into the input buffer on the die inside the IC package. Do all these different components connected together have 50 ohm impedance each?


If the frequency is high enough that these features can not be treated as lumped elements, then yes, they should be designed as near as possible to 50 ohm characteristic impedance. 

The traces in the IC package are routinely designed this way.

The ball of a BGA cannot practically be designed to have 50 ohm characteristic impedance, so we usually see some non-ideality (reflections) from the bga balls if the operating frequency is high enough. 

The bondwires between the package and the chip itself also can't practically be designed to have 50 ohm characteristic impedance. Therefore a wirebonded package also will have some non-ideal behavior due to the bond wires. Flip-chip packages don't require bondwires so they can help to reduce the return loss.