And will not love dare to trust itself in truth, And Love is taught hypocrisy from youth (1.72)
Byron thinks of youth as one of the only times in life (maybe the only time in life) when a person can be genuine. From that point on, the adult world gets a hold of you and pulls you into its realm of hypocrisy and double-dealing. This is all a shame because it prevents people from ever knowing genuine love and makes them obsessed with popularity and appearances.
His youth and features favoured the disguise, And should you ask how she, a Sultan's bride, Could risk or compass such strange phantasies, This I must leave sultanas to decide. (1.115)
Once Don Juan dresses up like a woman, he's able to fool most people because he's so young it's hard to tell he's a man. Being a younger boy means he doesn't have facial hair and his features are still smooth, so the point Byron is making here is that, the younger a person is, the more difficult it is to tell his or her gender.
It is enough that Fortune found him flush Of Youth, and Vigour, Beauty, and those things Which for an instant clip Enjoyment's wings. (10.5)
Don Juan is chockfull of youth and vigor, and these qualities often make other people jealous of him. To hear Byron tell the story, you'd swear that adults base 90% of their actions on the fact that they are jealous of those who are younger and more beautiful than them.
The favour of the Empress was agreeable; And though the duty waxed a little hard, Young people at his time of life should be able To come off handsomely in that regard. (10.22)
Don Juan becomes a favorite of Catherine the Great because of his youth and beauty. Byron hints here that he receives all kinds of gifts and favors as payment for "pleasing" Catherine in some way that Byron doesn't mention directly (psst: it might be sex).
Newton (that proverb of the mind), alas! Declared, with all his grand discoveries recent, That he himself felt only "like a youth Picking up shells by the great ocean—Truth" (7.5)
Byron admires Isaac Newton for being smart enough to admit his ignorance when it came to science. Byron doesn't like people who think they know everything, so he likes it when Newton compares himself to a young boy picking up shells by the ocean and looking for knowledge.
Suwarrow, who had small regard for tears, And not much sympathy for blood (7.69)
Compared to the young and innocent Don Juan, Suwarrow is an experienced military man who doesn't have much sympathy left in his body. It's pretty clear that Byron wants us to admire Don Juan's youth and sympathy by comparison.