The comments clarified your question a bit.
This question has nothing to do with entanglement.
A two particle wave function is $\Psi(x_1,x_2,t)$. None of your function is of that form. They are one particle wave functions.
The two particle wave function describes entangled particles when it cannot be written in the form
$$
\Psi(x_1,x_2,t)=\Psi_1(x_1,t)\Psi_2(x_2,t)\,.
$$
To the idea you had with your question: It is conceivable that you can smoothly paste together "two" wave functions $\Psi_1(x,t)$ and $\Psi_2(x,t)$ at some fixed $t$ but this will still give a single particle wave function because its space argument is one dimensional.
If you like here is my first version of the answer:
Doing calculations such as
\begin{align}
\frac{\partial\Psi_1}{\partial t_1}&=e^{-x_1}e^{i\sin(\pi t_1/3)}\cos(\pi t_1/3)\frac{i\pi}{3}-e^{-x_1^2}e^{i\cos(\pi t_1/5)}\sin(\pi t_1/5)\frac{i\pi}{5}\,,\\
\frac{\partial\Psi_1}{\partial x_1}&=-e^{-x_1}e^{i\sin(\pi t_1/3)}-2x_1e^{-x_1^2}e^{i\cos(\pi t_1/5)}\,,\\
\frac{\partial^2\Psi_1}{\partial x_1^2}&=e^{-x_1}e^{i\sin(\pi t_1/3)}-2e^{-x_1^2}e^{i\cos(\pi t_1/5)}+4x_1^2e^{-x_1^2}e^{i\cos(\pi t_1/5)}\,,\\
\end{align}
-and same for $\Psi_2$- one should be able to write down the Hamiltonians $H_1,H_2$
explicitly so that each $\Psi_1$ and $\Psi_2$ satisfies the Schrödinger equation
$$
i\hbar\frac{\partial \Psi_i}{\partial t_i}=H_i\Psi_i\,.
$$
Regarding your question what happens when we act $H_2$ on $\Psi_1\,:$ It is unlikely that
$$
i\hbar\frac{\partial \Psi_1}{\partial t_1}=H_2\Psi_i
$$
holds. In other words, it is unlikely that $H_2$ is the energy operator of "particle 1". The fact that $\Psi_1(\,.,t_1)$ and $\Psi_2(\,.,t_2)$ agree for two $t_1$ and $t_2$ is obviously totally irrelevant because the time evolution of $\Psi_1$ is governed by $H_1$ and not by $H_2\,.$
