Original proof of Uniform Boundedness Principle (Banach Steinhaus) and related questions

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.

See here: Alan D. Sokal, A really simple elementary proof of the uniform boundedness theorem, Amer. Math. Monthly 118, 450-452 (2011) https://arxiv.org/abs/1005.1585, http://dx.doi.org/10.4169/amer.math.monthly.118.05.450