What is the author's purpose in Act 2 lines starting with "Lysander riddles very prettily" and ending with "and good night, sweet friend" in A Midsummer Night's Dream?

Having lost his way to his aunt's house as he and Hermia pass through the forest in the dark, the young lovers stop in Act II, Scene 2 and decide to rest.  With this scene, Shakespeare introduces another aspect of love not previously mentioned in the play:  sexual attraction. When Lysander suggests that they lie together,

One turf shall serve as pillow for us both;
One heart, one bed, two bosoms, and one troth. (2.2.41-42)

Hermia becomes upset at the threat to her maidenhood, and she to "Lie further" away from her for her "innocence."  Lysander, then, explains that he meant nothing suggestive; he means that his heart lies with hers as they love each other and he is not lying about not doing anything ungentlemanly if he lies beside her:

 O, take the sense, sweet, of my innocence!
Love takes the meaning in love's conference.
I mean, that my heart unto yours is knit,
So that but one heart we can make of it;
Two bosoms interchained with an oath,
So then two bosoms and a single troth.
Then by your side no bed-room me deny,
For lying so, Hermia, I do not lie. (2.2.47-53)

But, Hermia still believes that he makes a riddle of the word lie. Still, she apologies if she misconstrues what he says; nevertheless, she prefers that Lysander sleep away from her as such a separation is appropriate for them.

Ironically, however, this physical separation to which Lysander agrees is the cause of Lysander's not seeing Hermia first when he awakens after Puck mistakenly dusts his eyes with the love potion intended for "the Athenian."  So, the dichotomy of dreams/reality in love is furthered in this second act with the ambiguity of feelings, the suggestion of sexual attraction, and as Lysander, so deeply in love with Hermia, then shortly finds himself infatuated with Helena, demonstrating, too, "what fools these mortals be" as Puck later comments. 

What is the author's purpose in Oberon's lines that begin "Thou hast mistaken" in A Midsummer Night's Dream, act 3, scene 2?

Whenever you are asked to analyse the importance of a specific quote or group of lines, the place to start is by reading the context of those lines: what happens before and after. This is going to be something that will help you work out the significance of those lines in particular. If we look at the lines you have highlighted, we can see that Oberon utters them to Puck just after they have observed Hermia and Demetrius and Helena and the chaos that has befallen them because of Puck confusing the situation further. Now both of the male Athenian lovers love Helena and none love Hermia. Oberon is rightly angry with Puck for having disobeyed him and (intentionally?) mistaking the Athenian males. Note the full quote of what he says to Puck:

What hast thou done? Thou hast mistaken quite
And laid the love juice on some true love's sight.
Of thy misprison must perforce ensure
Some true love turned, and not a false turned true.

Oberon is basically telling off Puck in these lines, saying that as a result of Puck's "misprison" or mistake, a true love has been altered rather than a "false love" being made "true." For me, the importance of these lines lies in the question that every director must answer. Did Puck mistake the Anthenian lovers deliberately? Is he the kind of character that likes chaos and mischief so much that he has intentionally made the situation more complex so that he can enjoy it? How would Oberon respond? In what tone of voice does he utter these lines? Thus this quote is very important in terms of the relationship beween Oberon and Puck and reveals a lot about Puck's character.