Most logicians would regard self-contradiction as a flaw, yes. The reason is that a good argument is supposed to be one whose conclusion must be true if its premises are true. If at some point in the argument a contradiction appears, then either (i) the reasoning was bad or (ii) the premises cannot all be true. That said, however, one can use possibilty (ii) to argue for something by what is called reductio ad absurdum: If you want to show that not-p, show that p (possibly together with other things that are agreed to be true) leads, via good reasoning, to contradiction. Then not all of the premises can be true. So if the ones other than p are, it isn't.

Now, how do you tell if you have a contradiction when the argument in words? There's no magic bullet, I'm afraid. Being able to translate into symbolic logic only helps so much, and in the really hard cases it'll be controversial how to do the translation, anyway. So, to a first approximation, you have reached a contradiction if you have reached a conclusion that cannot be true. There is more to say, to be sure, but most of it involves trying to explain what "cannot" means here.