Let $f:\Omega\to\mathbb{C}$ be a holomorphic function where $\Omega$ is an open subset of $\mathbb{C}$ such that $\{\frac{1}{z}:z\in\Omega\}$ contains a deleted neighborhood of $0$ in $\mathbb{C}$. I believe that $f:\Omega\to\mathbb{C}$ has a removable singularity at $\infty$ if and only if $z\to f\left(\frac{1}{z}\right)$ has a removable singularity at $0$. (Exercise!)
Let $f$ be the function in the first line of your question. We can write $f\left(\frac{1}{z}\right)=z+z^5$ for all $z\in\mathbb{C}\setminus \{0\}$. In particular, it is clear that $z\to f\left(\frac{1}{z}\right)$ has a removable singularity at $0$.
Let $f$ be the function in the fifth line of your question. We can write $f\left(\frac{1}{z}\right)=\frac{1+z^2}{1-z^2}$ for all $z\in\mathbb{C}\setminus \{0\}$. In particular, it is clear that $z\to f\left(\frac{1}{z}\right)$ has a removable singularity at $0$.
However, in this case $\lim_{z\to 0} f\left(\frac{1}{z}\right) = 1$. In particular, $f$ does not have a zero at $\infty$.
The following exercises are relevant to your question:
Exercise 1: Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function. If $f$ has a removable singularity at $\infty$, then prove that $f$ is contant. (Hint: Liouville's theorem.)
Exercise 2: Let $f:\Omega\to\mathbb{C}$ be a holomorphic function where $\Omega$ is an open subset of $\mathbb{C}$ such that $\{\frac{1}{z}:z\in\Omega\}$ contains a deleted neighborhood of $0$ in $\mathbb{C}$. We write that $f$ has a pole of order $m$ at $\infty$ if the holomorphic function $z\to f\left(\frac{1}{z}\right)$ (defined in a deleted neighborhood of $0$) has a pole of order $m$ at $0$. Prove that $f$ has a pole of order $m$ at $\infty$ if and only if there are complex numbers $c_1,\dots,c_m$ with $c_m\neq 0$ such that $f(z) - \sum_{k=1}^{m} c_kz^k$ has a removable singularity at $\infty$.
Exercise 3: Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function. If $f$ has a pole of order $m$ at $\infty$ (for some positive integer $m$), then prove that $f$ is a polynomial of degree $m$.
Exercise 4: Let $f:\Omega\to\mathbb{C}$ be a holomorphic function where $\Omega$ is an open subset of $\mathbb{C}$ such that $\{\frac{1}{z}:z\in\Omega\}$ contains a deleted neighborhood of $0$ in $\mathbb{C}$. Define what it means for $f$ to have an essential singularity at $\infty$.
Exercise 5: Prove that the entire function $f(z)=e^z$ has an essential singularity at $\infty$.
I hope this helps!