ἡ συμφορά, τῆς συμφορᾶς accident, misfortune, disaster

ὁ πρῶτος χορὸς προχωρεῖ 

See those dancing groups advancing one after the other, and imagine them, different colours, girls too, since we're not living in ancient Greece where women were most definitely not allowed to participate in official contests.
Revise ordinal numbers
   ἡ πρώτη ἡμέρα 
   ὁ δεύτερος χορός
   τὸ τρίτον μουσεῖον 
      καὶ τὰ λοιπά 
[ὑμνέω] ὑμνῶ, sing in praise
ὁ ὕμνος not hymn really, just song of praise
ᾄδω or ἀείδω (see Ὀδυσσεὺς καὶ ἡ Κίρκη lesson 9b) to sing (normally)
τὰ τοῦ Διονύσου ἔργα ᾄδων would be singing about his deeds, relating them
τὰ τοῦ Διονύσου ἔργα ὑμνῶν is praising them, singing about how great he is!
And that is precisely what ὁ πρῶτος χορός is doing.
Now decide πότερον ὑμνοῦμεν ἢ ᾄδομεν 
   country music?
   our national anthem?
πότερον ὑμνεῖ ἢ ᾄδει the national anthem our country's perfections?
I hope you have had no problem with πότερον ... ἢ .... meaning: (do we/you... )either .... or?

We left Dikaiopolis and family settling down outside the gates while others spent the night on the tiles. Are festivities over then?
Not quite. There are still contests to be seen, dancing contests as the picture shows us. Not in the temple, in the theater this time.
And grandad complains as usual, so many people, where can he sit?
Why does the priest make a libation?

The stage where the dancers perform is called ἡ ὀρχήστρα (dancing circle),
and 'to compete' is ἀγωνίζομαι from
   ὁ ἀγών, τοῦ ἀγῶνος lesson 4b
so: to agonize actually means to compete, to fight (death, for our lives)

τίνες ἀγωνίζονται;
   παῖδες ἀγωνίζονταί τε καὶ ἄνδρες
καὶ αἱ γυναῖκες; οὐκ ἀγωνίζονται καὶ αἱ γυναῖκες;