If I give my Gunslinger Sharpshooter a scope or a repeater, will that also affect his pistol attacks?
No. Weapon Mods do not affect secondary weapons.
Carried utility items (such as Talon or Blue Screen Rounds) however, do.
The one sort of caveat to this is when both of these are combined -- each time an enemy takes damage from a DoT (e.g., the fire caused by Dragon Rounds), the Execution ability of the Repeater can trigger, regardless of whether the fire was applied by the pistol or the Sniper shot.