$(1)$ 正確：因為 $\overset{\large\rightharpoonup}{AB} + \overset{\large\rightharpoonup}{BC} = \overset{\large\rightharpoonup}{AC}$，且 $\overset{\large\rightharpoonup}{AE} + \overset{\large\rightharpoonup}{EF} + \overset{\large\rightharpoonup}{FG} + \overset{\large\rightharpoonup}{GC} = \overset{\large\rightharpoonup}{AC}$，故正確。 $(2)$ 正確：在 $\triangle OAB$ 中，因為 $E, F$ 分別是 $\overline{OA}, \overline{OB}$ 的中點，所以 $\overline{EF} = \dfrac{1}{2}\overline{AB}$，且向量方向相同，即 $\overset{\large\rightharpoonup}{EF} = \dfrac{1}{2}\overset{\large\rightharpoonup}{AB}$，故 $\overset{\large\rightharpoonup}{AB} = 2\overset{\large\rightharpoonup}{EF}$。 $(3)$ 正確：因為 $\overset{\large\rightharpoonup}{DB} + \overset{\large\rightharpoonup}{BC} = \overset{\large\rightharpoonup}{DC}$，且 $\overset{\large\rightharpoonup}{DC} = \overset{\large\rightharpoonup}{AB}$，所以 $\overset{\large\rightharpoonup}{DB} + \overset{\large\rightharpoonup}{BC} = \overset{\large\rightharpoonup}{AB}$，即 $\overset{\large\rightharpoonup}{AB} - \overset{\large\rightharpoonup}{BC} = \overset{\large\rightharpoonup}{DB}$。 $(4)$ 正確：因為 $\overset{\large\rightharpoonup}{AB} + \overset{\large\rightharpoonup}{BF} + \overset{\large\rightharpoonup}{FE} = \overset{\large\rightharpoonup}{AE}$，且 $\overset{\large\rightharpoonup}{AE} = \overset{\large\rightharpoonup}{GC}$，所以 $\overset{\large\rightharpoonup}{AB} + \overset{\large\rightharpoonup}{BF} + \overset{\large\rightharpoonup}{FE} = \overset{\large\rightharpoonup}{GC}$。 $(5)$ 正確：因為 $\overset{\large\rightharpoonup}{AE} \mathbin{/\!/} \overset{\large\rightharpoonup}{OA}$，且 $\overset{\large\rightharpoonup}{BF} \mathbin{/\!/} \overset{\large\rightharpoonup}{BO}$，又正方形對角線 $\overline{OA}$ 與 $\overline{BO}$ 垂直，所以 $\overset{\large\rightharpoonup}{AE}$ 與 $\overset{\large\rightharpoonup}{BF}$ 配合垂直，即 $\overset{\large\rightharpoonup}{AE} \cdot \overset{\large\rightharpoonup}{BF} = 0$。