advantage of horn antenna
about private questions, I'll answer You privately.
About the difference between conical and piramidal... well, each one form has it's own property, but is the illumination requirements that set the needed form of the horn.
Basically, corrugated horns have lower cross-polarization and cleaner (say low sidelobe) beam than smoot walled horns.
The conical corrugated horn is used where a perfect (say almost perfect) pencil beam is required, while piramidal ones are historically used as standard gain horn.
The word "standard" means metrologic measurent instruments; since the gain is well predictable by geometric dimensions. In other words the standard horn appear as mechanically very simple to build but it came with a certificate, so it's a very expensive device.
Recentely the conical corrugated horn has been proposed to became a standard.
The advantage is that the corrugated path inside the horn is very, very, very low loss (say 0.001 to 0.01 dB/m) , so i'ts omhic losses need to not be known. Also conical corrugated has tipically an extreme low reflection coefiicient (i.e -30dB).
For requirement that are asking a beamwidth large in one dimension but small on the other one (i.e 90° x 1°) a special rectangular horn is used. It's called "sectorial horn" . The simple description is a piramidal horn with thin and long rectangular aperture.
Placing the long side vertically, You may broadcast a land whitout loose energy on the sky or on the ground.