Efficacy of the 10-20 montage for EEG
The placement in terms of location on the scalp determines what region of the brain is sampled. The electrode density determines how many locations are sampled. The spatial resolution of an EEG has been reported to be 7 mm (Im et al., 2006), meaning the variability in determined source and verified source (via fMRI in the study cited) is approximately 7 mm. Hence, increasing the density below 7 mm may not be too helpful. The 10-20 system utilizes an inter-electrode distance of 60 - 65 mm (American Clinical Neurophysiology, 2006), indicating it may make sense to use more electrodes when high resolution is needed. Note, however, that MRI and MEG have better spatial resolution than EEG and are methods of choice when you wish to do accurate source localization. EEG has excellent properties in the temporal domain, and a combination of MEG/EEG has great characteristics in both spatial and temporal domains.
Standardized EEG montages indeed promote uniformity so that different studies can be compared more efficiently.  
Scrubbing with electrode paste primarily reduces impedance (increases conductivity). Hair and the skin don't conduct well and by applying a conductive paste impedances are substantially reduced. Mostly the technicians also scrape a little bit to ensure even better conductance by removing the fatty top layer of the skin. Also important is that differences in electrode impedances negatively impact recordings (American Clinical Neurophysiology, 2006). Impedance matching is hence necessary and using paste aids this purpose. So yes, conductive paste is essential.


Reference
- Im et al., J Neurosci Methods (2007); 161(1): 142–54