First of all, in order to have $(iii)$ be equivalent to $(i)$ and $(ii)$, we need that $Y$ is complete (a Banach space) too.
As a counterexample when $Y$ is not complete, consider $X = \ell^p$, and $Y$ the subspace of sequences with only finitely many nonzero terms. Then let
$$(T_nx)_k = \begin{cases}x_k &, k \leqslant n\\ 0 &, k > n. \end{cases}$$
$T_n$ is a bounded sequence of linear operators $X\to Y$, and $T_n(x)$ converges for all $x$ in the dense subspace $Y$ of $X$. But $T_n(x)$ does converge only for $x\in Y$, so neither $(i)$ nor $(ii)$ hold in this example.
Your attempt to prove $(i) \Rightarrow (iii)$ does not work, there are two problems. First, from $\lVert T_n(x)\rVert < N$, you cannot deduce $\lVert T_n\rVert_{op}\lVert x\rVert < N$, since you only have the inequality $\lVert T_n(x)\rVert \leqslant \lVert T_n\rVert_{op}\lVert x\rVert$, not equality.
Second, the bound on the sequence $T_n(x)$ depends on $x$, you only have $\lVert T_n(x)\rVert \leqslant N(x)$.
To show the uniform boundedness, you consider the closed sets $$A_K = \{x \in X : \lVert T_n(x)\rVert \leqslant K \text{ for all } n\}.$$
Since $T_n(x)$ is bounded for every $x\in X$, you have
$$X = \bigcup_{k=1}^\infty A_k.$$
From Baire's theorem, deduce that some $A_k$ is a neighbourhood of $0$ in $X$.
For the implication $(iii) \Rightarrow (ii)$ (under the assumption that $Y$ is complete!), note that
- the set $C = \left\{ x \in X : \lim\limits_{n\to\infty} T_n(x)\text{ exists}\right\}$ is a linear subspace of $X$,
- the map $T\colon C \to Y$ defined by $T(x) = \lim\limits_{n\to\infty} T_n(x)$ is linear and continuous.
Then the general theory asserts the existence of a continuous (linear) extension $\tilde{T}\colon \overline{C}\to Y$. Since $C$ is by assumption dense, $\overline{C} = X$. Now use the boundedness of the norms - the equicontinuity of the family $(T_n)$ - to deduce that actually $C = X$.
Best Answer
Well, it is a good idea to look at the Wikipedia page first. There you'll find a link to the original article by Banach-Steinhaus which was motivated by the theory of Fourier series.
Concerning your question whether it was standard to apply the Baire category theorem at those times, very much so. The Polish school (e.g. Sierpinski, Lusin, Souslin, and of course Banach) applied it routinely, just look at their papers in the early volumes in Fundamenta Mathematicae.
Lastly, the proof in that paper you link to is quite a common gliding hump argument, which was recently published by Sokal in the Monthly. You'll find further historical remarks and discussion in his article.