As far as your added question goes, no, I'm afraid your thought process is invalid. You are essentially using a classic misapplication of the Divergence Test.
Divergence Test. If the limit of $a_n$ as $n\to\infty$ is not equal to $0$ (either does not exist, or exists and is not equal to $0$), then the series
$$\sum a_n$$
diverges.
(Sometimes the Divergence Test is phrased in the contrapositive:
If
$$\sum a_n$$
converges, then $\lim\limits_{n\to\infty}a_n = 0$.
but you are still trying to use it by affirming the consequent, which is a logical fallacy and an invalid mathematical argument). The Divergence Test can never let you conclude a series converges (look at the name: it's a dead giveaway). But that's what you are trying to do.
To go over the points I made in comments: whenever you have a series,
$$\sum_{n=1}^{\infty}a_n$$
you automatically get two sequences that are associated to the series. The sequence that you are actually interested in, vis a vis the series, is the sequence of partial sums:
$$\begin{align*}
s_1 &= a_1\\
s_2 &= a_1+a_2\\
s_3 &= a_1+a_2+a_3\\
&\vdots\\
s_n &= a_1 + a_2 + \cdots + a_n\\
&\vdots
\end{align*}$$
When we ask whether a series $\sum a_n$ converges, we are really asking whether the sequence $\{s_n\}$ of partial sums converges. When we say that the series "converges to $L$", $\sum a_n = L$, we are really saying that the sequence $s_n$ converges to $L$. When we say the series $\sum a_n$ diverges, we are really saying that the sequence of partial sums $\{s_n\}$ diverges.
Unfortunately, the sequence of partial sums is usually very hard to get a hold of. Thing about even a very simple series, like the harmonic series,
$$\sum_{n=1}^{\infty}\frac{1}{n} = 1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4}+\cdots + \frac{1}{n} + \cdots$$
The sequence of partial sums is pretty complicated, in terms of trying to get a formula for the $n$th term:
$$\begin{align*}
s_1 & = 1\\
s_2 &= 1 + \frac{1}{2} = \frac{3}{2}\\
s_3 &= 1 + \frac{1}{2}+\frac{1}{3} = \frac{11}{6}\\
s_4 &= 1 + \frac{1}{2}+\frac{1}{3}+\frac{1}{4} = \frac{25}{12}\\
&\vdots
\end{align*}$$
The consequence of this difficulty is that we usually have a very hard time telling directly whether the sequence of partial sums converges, let alone what it converges to; because it's so hard to get a hold of the series.
So instead we look at the other sequence that we get from the series (which I alluded to above): when we have the series
$$\sum_{n=1}^{\infty}a_n$$
we also get the sequence of terms,
$$a_1,a_2,a_3,\ldots,a_n,\ldots$$
This is a much easier sequence to get a hold of, because we usually have it right in front of us.
Now, the sequence of terms is not what we are actually intereted in (we are actually interested in the sequence of partial sums), though it is of course related to the sequence of partial sums. Because the sequence of terms is so much easier to handle, though, we would like to be able to infer properties of the sequence of partial sums (in particular, whether it converges or not) from properties of the sequence of terms.
This is what pretty much all the tests for series that you see in Calculus II are about: they are ways in which you may be able to infer whether the sequence of partial sums converges or not, based on properties of the sequence of terms.
But, ultimately, we are looking in the wrong place: we are like the drunk searching for his keys under the streetlight, not because that's where the keys were lost, but because that's where there is enough light. Because we are really looking at the wrong place (and merely hoping to catch a glimpse of what we are really looking for from the corner of our eye), almost all tests are limited in some way: either you cannot always use them (e.g., the limit comparison test cannot be used if the series have positive and negative terms; the alternating series test cannot be used if the series is not alternating, etc); or they are not always conclusive (the Divergence Test can tell you a series diverges, but it can never tell you a series converges; the Ratio Test can be inconclusive; the Alternating Series Test can be inconclusive; etc). Because, ultimately, we are focusing on the wrong thing.
Which is what makes things so complicated and confusing for students. There is no recipe; instead, there are just a bunch of things you can try, and which may or may not work. You need to try several things, and you need to remember exactly what you can conclude and what you cannot from the different things you try, and when you can use them and when you cannot. It's a lot to keep in mind, but unfortunately it's the best we can do.
With practice, one comes to recognize certain series, to get more series "under one's belt" (so that we can do things like apply Comparison or Limit Comparison), or through experience get a feel for what kind of tests tend to work well with what kind of series (for instance, my experience tells me that trying to use the Ratio Test with the series you give in your addendum would be a waste of time, since it would come out inconclusive). But I'm afraid that comes with experience, and there is no clever mnemonic that you can memorize, or clever trick you can always use, or magic elixir you can drink, that will let you do it easily and every time (and without having to think about it too much). There just isn't any such thing, just like there isn't any such thing for finding antiderivatives.
For the series
$$\sum_{n=2}^{\infty}\frac{1}{n^2-1},$$
what you did was apply the Divergence Test to see if the series might diverge; this is generally a good first step, because it is generally easy to do, and if the terms fail the divergence test (the terms do not go to $0$), then you are done: the series diverges. You performed the test correctly: it is indeed the case that
$$\lim_{n\to\infty}\frac{1}{n^2-1} = 0$$
because the denominators grow without bound. Unfortunately, what this means is that the terms pass the Divergence Test, and therefore that the divergence test does not settle whether the series converges or diverges: the test is inconclusive.
Instead, you might note that this series is very close to the series $\sum\frac{1}{n^2}$; so you might want to try a comparison or limit comparison test with that series (assuming you know whether $\sum\frac{1}{n^2}$ converges or diverges).
