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ἐν ᾧ δὲ ἄπεισιν ἥ τε Μυρρίνη καὶ ἡ Μέλιττα....

ἐν ᾧ δέ while (temporal while = during the time that) not to be confused with our:
[while] X does this, Y [by contrast] does that
       ὁ μὲν Α πονεῖ, ὁ δὲ Β καθεύδει
ἄπεισιν they are away, not present. From ἀπό (away from) and εἰσίν they are
the slave is not here ὁ δοῦλος ἄπεστιν
while I'm not here.... ἐν ᾧ δε ἄπειμι
See conjugation of εἰμί and compounds πάρειμι, (I'm present) and ἄπειμι, (I'm absent, away) here.
 

ὁ μὲν πάππος πονεῖ ἐν τῷ κήπῳ.

Grandfather in garden. Visualize him, think of a garden and call it κῆπος, κῆπος μέγας, κῆπος καλός,
    κῆπος μέγας - πόνος πολύς, ....
ὁ πάππος: have you still got one: (ἆρα) ἔτι ἔχεις πάππον; or, better:
     ἔτι εστί σοι πάππος;
So now the family has grown: ὁ πάππος, ὁ πατήρ, ἡ μήτηρ, ὁ υἱός, ἡ θυγάτηρ, ὁ/ἡ παῖς and the dog, ὁ κύων, τοῦ κυνός.
 

ὁ δὲ Φίλιππος καὶ ὁ Ἄργος βαδίζουσιν πρὸς τὸ αὔλιον. That's where τὰ πρόβατα (the sheep, next sentence) are kept.

Now we have places where we work and live:
ὁ ἀγρός, ὁ κλῆρος, ὁ κῆπος, ὁ οἶκος or ἡ οἰκία and where sheep live: τὸ αὔλιον
 

Description of the dog:

ὁ Ἄργος κύων ἐστὶ μέγας τε καὶ ἰσχυρός, I suppose we could add τε καὶ ἀνδρεῖος because of what he does:
τήν τ' οἰκίαν φυλάττει καὶ τὰ πρόβατα.
What is this τ' ? The ε of τε is dropped before a vowel/diphthong sound. Why? as usual, for ease of pronunciation. That's why we say: doesn't, don't, won't shan't. It's called elision (see p. 58 in your book).
φυλάττω I guard. Think of soldiers guarding places, of dogs, of dragons guarding treasures and call their action φυλάττουσιν, φυλάττει.
What do we guard? τί φυλάττομεν; καὶ σύ, τί φυλάττεις;
 

What do they see while they are walking? Where are they walking to? So where is the αὔλιον τε καὶ τὰ πρόβατα?

ἐν ᾧ δὲ βαδίζουσιν ἀνὰ τὴν ὁδόν up the road (the αὔλιον must therefore be in the hills)
ὁ Φίλιππος λαγών ὁρᾷ ἐν τῷ ἀγρῷ τρέχοντα.
Philip sees (ὁράει shortens to ὁρᾷ, the ι disappears as a real letter and becomes subscript to show that there was an ι originally. We've seen this already in the picture.)
 

So what does he do then?

λύει οὖν τὸν κύνα
He lets the dog off the chain/leash and sicks him on saying, like in the picture: ἴθι δή, Ἄργε, δίωκε.
 

And then, what does the dog do, (a) much to Philip's joy? (b) to his despair?
 

• (a) ὁ κύων ὑλακτεῖ (ὑλακτέ-ε) καὶ διώκει τὸν λαγών.
  Remember:
  διώκει  the dog does something, it gives chase
  δίωκε! Philip gives it an order: go on, give chase!
  Dogs have a tendency to bark: ὑλακτοῦσιν πολύ. Especially at night, that's when we tell our dog: μὴ ὑλάκτει (don't go on barking, that disturbs the neighbours). Except that here in Portugal people seem to be deaf, dogs go on barking all through the night, under their owners' windows and nobody seems to notice:
  οἱ μὲν κύνες ὑλακτοῦσιν, πολλάκις ὑλακτοῦσιν, ἀεὶ ὑλακτοῦσιν, οἱ δὲ ἄνθρωποι οὐ λέγουσιν· μὴ ὑλακτεῖτε (ὑλακτέ-ετε), ἥσυχοι ἔστε. And while the Portuguese καθεύδουσιν, οἱ δὲ ξένοι (foreigners) οὐχ ἡσυχάζουσιν διότι οἱ κύνες ἥσυχοι οὐκ εἰσίν.
   
• (b) The dog obeyed, gave chase. What the boy, obviously not over-intelligent, hadn't realised: ὁ λαγώς φεύγει. That's what hares do, φεῦγουσιν, ἀνδρεῖοι γὰρ οὐκ εἰσίν.
  Where to? ἀνὰ τὸ ὄρος.
  They were walking up the road ἀνὰ τὴν ὁδόν
  dog and hare are running: τρέχουσιν ἀνὰ τὸ ὄρος
  ἐν ᾧ δὲ βαδίζει ὁ παῖς, ὁ μὲν λαγὼν τρέχει. καὶ ὁ κύων τρέχει.
   

ἆρα βραδέως τρέχει ὁ λαγώς; οὐ, οὐ βραδέως, ἀλλὰ ταχέως τρέχει.
οὕτω ταχέως τρέχουσιν, ὥστε ὁ Φίλιππος δι' ὀλίγου οὐκέτι ὁρᾷ (ὁρά-ει) αὐτούς.

δι' ὀλίγου soon, after a short while. Another elision, this time of the α of διά. Easier to say δι' ὀλίγου than διὰ ὀλίγου.
Because hare and dog run so fast, it is soon impossible to see them:
διότι οὕτω ταχέως τρέχουσιν, δι' ὀλίγου οὐ δυνατόν ἐστι ὁρᾶν αὐτούς, (ὁρά-ειν, itself a contraction of ὁρά-ε-εν)
 

οὐ δυνατόν ἐστι ὁρᾶν οὔτε τὸν λαγών, οὔτε τὸν κύνα.

οὔτε .... οὔτε: οὐ (not) + τε (and) are joined together as οὔτε. οὐ doesn't normally carry an accent, except when followed by a weak enclitic (that in-clining leans on it en-clitic. And that is exactly what this weak τε [and] is)
(my cat) neither in the house, nor in the garden: οὔτε ἐν τῷ οἴκῳ, οὔτε ἐν τῷ κήπῳ
(Xanthias) neither in the field, nor in the road: οὔτε ἐν τῷ ἀγρῷ, οὔτε ἐν τῇ ὁδῷ