How to punctuate a sequence of steps in mathematical text?

I'm really glad that the top example bothers you - it bothers me too. Unfortunately I think that we are in a fairly small minority. Many people seem to think it's OK (maybe, even, preferable) to write mathematics in an impenetrable way. Perhaps I am just way too cynical, but I sometimes think this is an instance of the syllogism

you don't understand what I wrote, therefore I am smarter than you

which is surely not valid. Those of us who believe that clear writing and thorough explanations are vital parts of mathematics just need to keep on setting an example against this kind of thing.

Anyway... <end of rant>... back to the point... IMHO the way you put it at the end is definitely better than the first, though possibly even better still would be to use words: $$x^2-4=0\ ,\quad\hbox{so}\quad (x+2)(x-2)=0\quad\hbox{and therefore}\quad x=2\ \hbox{or}\ x=-2\ .$$ Actually, despite what I said above, I could possibly be convinced that this is overkill - after all, it is a very simple example. On the other hand, anyone who neglects or refuses to write properly in simple cases is unlikely to explain themselves clearly in more extended contexts where the writing becomes even more important.

I would also suggest that the set notation in your final line $x\in\{2,-2\}$ is rather unnecessary, and slightly obscures the point that we are doing algebra here (though once again, this is a simple example - I might not say the same in a different case). Finally, the three parts of your statement are actually all equivalent (the logic goes both ways, not just left to right), and it might be good to make this clear, if it is something that's important for your argument.

Hang in there, keep writing as well as you can, and don't let anyone tell you that you're wasting your time! Good luck!