They both won for the best tragedy multiple times at the "Great Dionysia", a festival dedicated to Dionysos during wich a theater contest was held. So they were famous, and remained as such during the following centuries, enabling the conservation of at least a part of their work. The attendance to the festival was mandatory to every Athenian citizen, so the concept of a critic was unnecessary, as they all attended the premiere. Sadly, I don't know how the winner of the contest was chosen.

One of the best ways to learn about how people felt about the great tragedians of ancient Athens is to look at the great comics. Although Sophocles, Aeschylus and many other playwrights of the day were often lampooned quite harshly, the comic writers often citing very specific information in order to make their caricatures, Euripides was particularly lampooned for being aloof and overly intellectual.

I'm sorry I can't answer your question of whether there were professional critics whose sole job was to evaluate the works submitted during the Dionysia or the Lenaia, but seeing as how these festivals only lasted for short amounts of time every year, its hard to see someone in an agrarian society being solely devoted to art criticism. More than likely there were well known citizens whose opinions on tragedies were well respected, but primarily made their livings in other ways.

Oh I just saw this. In essence, no, not really. That being said all early Greek poetry had a competitive element and Athenian drama literally was written for and acted at a competition (well, a religious festival). The judges were simply chosen from amongst the citizenry (and there's a whole welter of complex socio-cultural things going on there with what makes good poetry and who gets to judge etc).

As to how they were received well we have a number of ways of answering this. One is imitation and intertextuality with other writers, another is frank discussion as in Plato and Xenophon. However we also possess epigraphic lists like the didaskaliai and nikai which list people competing, names of actors and what place each play came.

If you have time I'd suggest something like Sommerstein's "Greek Drama and Dramatists" which is written at a very introductory level and I think covers these sorts of questions.

I'm not familiar with the immediate response beyond what has already been stated. However, I think Aristotle's Poetics merits mention here. It has a claim to be considered the foundational text of literary criticism and aesthetics. Aristotle devotes much of his time to examining what constitutes a good and a poor play (or epic). He also includes numerous examples of the concepts as he explains them. I do not have the text in front of me, but I do remember that, predictably, Homer is taken as the gold standard on most concepts (such as plot, poetic skill, etc).