As any companion can wear any piece of gear, is there a reason not to give them the heaviest armor and the biggest guns, regardless of their "class" or personality? I did not see weapon-specific skills on the character page, thus I assume that the default value is used to calculate damage.

Only played all the way through once, but I think your assumption is correct.

Well they seem to get bonuses if you equip a weapon matching what they do for their specials or the weapon type they originally used.  The thing is that if you level up damage over all becomes very over powered so they will do fine with any weapon you wish to throw on to them.  I do prefer to use the heaviest armor for my companions as I feel the skill bonuses on the lighter armors are wasted on them and the heavier stuff really helps their ability to survive battles.

Nothing stopping you, but it's pretty much unnecessary. Even with just a little bit of tinkering then are pretty hardcore

Played through the whole game three times in hard difficulty. Highest armor + highest DPS is what I settled upon by the end of the my first play through. Didn't change for subsequent playthroughs as I had no role-playing reason to do so. I didn't notice any advantage in sticking to any different armor or weapon type either. I only noticed my companions  more often.