For instance ‘Trooper’ which gives assault rifle proficiency with +20% damage and accuracy.
Do you only get those stat improvements when using a rifle?
Similarly ‘Sniperist’ gives sniper rifle proficiency +25% damage and -4 willpower. Do you only gain the damage and lose the willpower if you equip a sniper rifle?
Yes.
Yes to the first part, no to the second part.
Wow. Can’t really see any situation where I’d want to take the weapon specific perks with negative elements then.
Actually, plenty of times - if not always 
It allows for very overpowered builds. For example, strongman (+30% damage heavy weapons, - 10 perception) for an assault/sniper with dash, quickaim and rapid clearance armed with a heavy machine gun is absolutely devastating.