Splinter weapon does not "hit" adjacent foes. It only does armor ignoring damage to adjacent foes. [anthem of flame] and [go for the eyes] effect target foes not adjacent. So the answer is no target foe only.
Hey I was wondering if I use Splinter Weapon + a skill like Anthem of Flame, will it only make the target foe burn or all adjacent foes as well?
Same thing with GftE, will the critical hit only be on the target foe?
splinter weapon isn't actually damage from your weapon, so no. Think of it more like ancestor's rage, except it triggers whenever you hit a foe rather than used on an ally. Basically, the only thing you can use with splinter weapon to augment it's damage is By Ural's Hammer.