The different attack-types (Shell, laser, minor electric, missile, etc) have no actual effect on damage dealt out. Their only purpose (AFAIK) is to give generic types that certain ships are immune to (Bulletproofs against shells, Eye Bots against missiles, etc).
The strong against part of the tooltip again doesn't mean the attack-type, it means the attack of that *particular* unit.
Yes, the base stats such as health, range, reload time, shields, damage amount are the main factors, but each ship has multipliers or divisors on their damage against each other unit type which can (Massively) skew engagement results one way or the other.
Mark I Fighters beat Mark I Bombers because the fighters have a massive damage bonus against bombers of the same mark level. And bombers have (I think) a massive nerf against fighters. Part of the core-unit RPS triangle: Fighters beat bombers, which beat frigates, which beat fighters.
Hardened targets are the big emplacements, Force Field generators, Fortresses, Superfortresses, that kind of thing.
Hope this clarifies somewhat.