It seems that weapon bonuses that are on the weapon itself are accounted for in the weapon's stats, but are bonuses for skills, class mods, badass bonuses, etc? If not, is there any way I can see what my actual stats are?
Weapon stats shown are always the base stats, not including skills, mods or badass bonuses.
To test this I checked out an SMG. With no bonuses, no skills or class mod, it has 54 damage. With a class mod increasing SMG damage 71%, a skill increasing bullet damage 20% and about 3% damage boost from Badass stat bonuses. The gun still showed 54 damage.
Note gun damage also doesn't indicate your level's effect on damage, so guns generally do vastly more real damage than their stats indicate; stats are mostly there to compare guns.