Are input filters always required for switching power supplies?
Is my analysis of the source of this EMC problem correct? (SMPS
  generating input noise due to high load).


I don't know if this is the cause of your 'loud GSM buzz' but it is above the compliance limit, and your tests indicate that it is coming from the SMPS. The peak conducted noise at 150kHz is ~90dB&mu;V or ~30mV, which is about what I would expect from a switching regulator with no input filter.   


  Are input filters always necessary in order to reduce SMPS switching
  noise


Yes, if the noise is unacceptable. A buck switching regulator draws current in pulses, so it will always generate noise at the input. The 100&mu;F electrolytic capacitor in your circuit must be providing some filtering, but apparently not enough. Using a larger capacitor probably won't help much because its inductance will likely be higher, which could make it even less effective. 

The usual way to tame EMI is to use an LC filter. Texas Instruments AN-2155 (Layout Tips for EMI Reduction in DC/DC
Converters) includes this circuit:-



As well the 'bulk' filter capacitor CIN6, several smaller capacitors are used to reduce impedance at increasingly higher frequencies. L2 and CIN7 form a low pass filter which reduces noise current conducted back into the power supply. Noise voltage may still be quite high at the SMPS, but the LC filter takes care of it. This filter should be dedicated to the SMPS, with other circuits fed from the supply side.     

LC filters will resonate if they have high Q, which may be a problem if the resonant frequency is close to the switching frequency. Therefore the capacitor values may need to be 'tuned' for best results. The bulk capacitor's ESR can also be utilized to reduce filter Q. 


  or can this problem be solved just by improving the PCB layout?


Probably not by itself, but a bad layout will exacerbate the problem. Your layout looks OK to me, except I would put a 1&mu;F ceramic capacitor close to the VIN pin. I would also remove the copper pour from areas where it isn't doing any good (such as under the inductor) and put lots of vias through it where I want to augment the ground plane.