The presence of extreme horizontal branch (EHB) and blue hook stars in some Galactic globular clusters (GGCs) constitutes one of the remaining mysteries of stellar evolution. While several evolutionary scenarios have been proposed to explain the characteristics of this peculiar population of evolved stars, their observational verification has been limited by the availability of spectroscopic data for a statistically significant sample of such objects in any single GGC. We recently launched the SHOTGLAS project with the aim of providing a comprehensive picture of this intriguing stellar population in terms of spectroscopic properties for all readily accessible GGCs hosting an EHB. In this first paper, we focus on ω Cen, a peculiar, massive GGC that hosts multiple stellar populations. We use non-LTE model atmospheres to derive atmospheric parameters (Teff, log g and N(He)/N(H)) and spectroscopic masses for 152 EHB stars in the cluster. This constitutes the largest spectroscopic sample of EHB stars ever analyzed in a GGC and represents ≈20% of the EHB population of ω Cen. We also search for close binaries among these stars based on radial velocity variations. Our results show that the EHB population of ω Cen is divided into three spectroscopic groups that are very distinct in the Teff − helium abundance plane. The coolest sdB-type stars (Teff ≲ 30 000 K) have a hydrogen-rich atmosphere, populate the theoretical EHB region in the Teff − log g plane, and form 26% of our sample. The hottest sdO-type stars (Teff ≳ 42 000 K) make up 10% of the sample, have a hydrogen-rich atmosphere and are thought to be in a post-EHB evolutionary phase. The majority of our sample is found at intermediate temperatures and consists of sdOB stars that have roughly solar or super-solar atmospheric helium abundances. It is these objects that constitute the blue hook at V > 18.5 mag in the ω Cen color-magnitude diagram. Interestingly, the helium-enriched sdOBs do not have a significant counterpart population in the Galactic field, indicating that their formation is dependent on the particular environment found in ω Cen and other select GGCs. Another major difference between the EHB stars in ω Cen and the field is the fraction of close binaries. From our radial velocity survey we identify two binary candidates, however no orbital solutions could be determined. We estimate an EHB close binary fraction of ≈5% in ω Cen. This low fraction is in line with findings for other GGCs, but in sharp contrast to the situation in the field, where around 50% of the sdB stars reside in close binaries. Finally, the mass distribution derived is very similar for all three spectroscopic groups, however the average mass (0.38 M⊙) is lower than that expected from stellar evolution theory. While this mass conundrum has previously been noted for EHB stars in ω Cen, it so far appears to be unique to that cluster.