①$2\otimes 5=2^{2}-2\times 2\times 5=-16$，故①正确；

②$\sqrt [3] {2\otimes \left(-1\right)}=\sqrt [3] {2^{2}-2\times 2\times \left(-1\right)}=\sqrt [3] {0}=0$是有理数，故②错误；

③方程$x\otimes y=0$得$x^{2}-2xy=0$是二元二次方程，故③正确：

④不等式组$\left\{\begin{array}{l}\left(-3\right)\otimes x+1 \gt 0\\2\otimes x-5 \gt 0\end{array}\right.$等价于$\left\{\begin{array}{l}\left(-3\right)^{2}-2\times \left(-3\right)x+1 \gt 0\\2^{2}-2\times 2x-5 \gt 0\end{array}\right.$，

解得$-\dfrac{5}{3} \lt x \lt -\dfrac{1}{4}$，故④正确；

故答案为：①③④.

①$2\otimes 5=2^{2}-2\times 2\times 5=-16$，故①正确；

②$\sqrt [3] {2\otimes \left(-1\right)}=\sqrt [3] {2^{2}-2\times 2\times \left(-1\right)}=\sqrt [3] {0}=0$是有理数，故②错误；

③方程$x\otimes y=0$得$x^{2}-2xy=0$是二元二次方程，故③正确：

④不等式组$\left\{\begin{array}{l}\left(-3\right)\otimes x+1 \gt 0\\2\otimes x-5 \gt 0\end{array}\right.$等价于$\left\{\begin{array}{l}\left(-3\right)^{2}-2\times \left(-3\right)x+1 \gt 0\\2^{2}-2\times 2x-5 \gt 0\end{array}\right.$，

解得$-\dfrac{5}{3} \lt x \lt -\dfrac{1}{4}$，故④正确；

故答案为：①③④.