The chain is actually transient.
First, let me address your question about the sum: because of the transience of the chain, if you correctly compute $\mathbb E[T_0 \mid X_0 = 0]$ you would apparently get $\infty$. This will be a disappointing result, of course, because it won't settle the original question; you will only know that the chain is not positive recurrent, but it could still be either null recurrent or transient.
Proof of transience: There is precisely one path that starts at $0$, leaves, and does not return; it's the path for which all steps are to the right. The probability of this path is equal to
$$\prod_{i=0}^{\infty}\frac{i^2 + 1}{i^2 + 2},$$
whence recurrence is equivalent to that product being $0$. We'll check by turning the product into a sum via a logarithm:
$$\log \left(\prod_{i=0}^{\infty}\frac{i^2 + 1}{i^2 + 2} \right) = \sum_{i=0}^{\infty} \log \left(\frac{i^2 + 1}{i^2 + 2} \right) \tag{$\bigstar$}$$
I claim the sum converges to a negative number instead of diverging to $-\infty$, as follows. If we negate the terms we have
$$\log \left( \frac{i^2 + 2}{i^2 + 1} \right) = \log \left(1 + \frac{1}{i^2 + 1} \right).$$
Note that for $x > 0$, we have $\log(1 + x) < x$; hence, we have
$$\log \left( \frac{i^2 + 2}{i^2 + 1} \right) < \frac{1}{i^2 + 1}$$
and since the right side is summable, so also is the left.
This implies that the sum in ($\bigstar$) converges to a finite negative number (instead of diverging to $-\infty$), so the the infinite product above it is therefore positive, and the chain is transient.
Here's another way to see the transience of the chain. I'm not sure if it's cleaner either conceptually or computationally; I'll leave that decision to the reader.
Consider a chain started at state $n \neq 0$, and for let $A_k$ denote the event that the walk takes exactly $k$ steps to the right (from state $n$) before then taking one step back to $0$. Clearly $\mathbb P(A_0) = \frac{1}{n^2 + 2}$, and you can verify that for $k \geq 1$,
$$\mathbb P(A_k) = \left[\prod_{i=0}^{k-1} \frac{(n+i)^2+1}{(n+i)^2 + 2}\right] \frac{1}{(n+k)^2+2} \leq \frac{1}{(n+k)^2 + 2}.$$
Note that $\{T_0 < \infty\}$ can be expressed as a disjoint union $ \cup_{k=0}^{\infty} A_k$, so
$$\mathbb P(T_0 < \infty \mid X_0 = n) = \sum_{k=0}^{\infty} \mathbb P (A_k) \leq \sum_{k=0}^{\infty} \frac{1}{(n+k)^2 + 2} = \sum_{k=n}^{\infty} \frac{1}{k^2 + 2}. \tag{$\bigstar \bigstar$}$$
Since the full series
$$\sum_{k=0}^{\infty} \frac{1}{k^2 + 2}$$
converges, this implies that the rightmost side of ($\bigstar \bigstar$) can be made arbitrarily small; in particular, it can be made less than $1$. This implies that there is some $n$ for which a chain started there has a probability less than $1$ of ever returning to $0$, which proves the transience of the chain (since it is irreducible).
