The conversion launcher uses seagull missiles.
Shotguns are weaker vs armored enemies, that is true. But they murder everything else.
In this mod, shotguns have ridiculous high accuracy compared to the other weapons. This is especially useful for the early game when your troops can't hit shit. Everyone can hit something with a shotgun. Seriously, this is a shotgun mod. You will understand once you get your hands on a mammoth chain shotgun.
The higher damage shotguns (combat, heavy, blunderbus, CAWS) are still ok for medium armored stuff (~25 armor). You should know from experience that a gal in warrior armor (30 armor) still takes damage from shotguns. For tougher armor, the AP ammo of military shotgun and CAWS are still better than AK rifles. You can also carry a backup weapon and/or use explosives. On top of that, mix your squad. Half shotguns, other half anti armor/long range.
The high damage pistols are excellent backup weapons (heavy pistol, magnum). Especially the magnum pairs well with a shotgun, as it has power AND range. Shotgun and magnum is probably the best early game weapon combo. Javelins and magnum is equally strong on a skilled gal.
What shotguns to use? Heavy for gals that have low firing skill, combat/trench gun for those with better firing.
I've looked at your memorial to see from which enemies your gals died. A lot died from blood hounds, dogs and ratmen. Shotguns are especially great vs these enemies.
Considering rifles: The early game rifles are really the weakest weapon type in this mod. They are only good for long range shooting (30+ tiles). Below 30 tile distance the pistol's aimed shot is better, as the typical pistol can do two aimed shots in the time a rifle shoots once. Pistols are also better at snap shots, and smgs have way better autofire than rifles.
Rifles are then best used to make something bleed from a long distance. A bleeding enemy will die eventually. So use rifles in a situation where you can wait until that happens. The RFC carbine is considered the best early game rifle for this tactic. Fast and accurate aimed shot, high armor penetration value. Not much damage though.
The rifles that are actually good in any situation are higher up the tech tree. Lasgun and smartrifle are what you are looking for.
Tl;dr: In the early game trenchgun + magnum is the best weapon loadout, next tier is smartrifle.