I don't really understand what the "strength requirement" for weapons influences. I can equip guns even when they have a requirement well above my character's strength (my character's strength is 1, so this is pretty much every weapon in the game). They are all quite accurate, both in and out of VATS. Does it directly govern damage even for non-melee weapons, or what?

Accuracy for non melee weapons, how fast you use them for melee weapons.

Think that if you don't meet the requirements of the weapon - strength and the "hidden" gun skill requirement, the weapon won't do optimal damage.