There are quite a few questions being asked here, and all of the answers can be found in chapter three. I will answer the first two questions, since they are related to each other. About halfway through the chapter, Tom is on his way home after playing games with other kids in the area. He comes to Jeff Thatcher's house, and he sees a girl that he is immediately smitten with.

As he was passing by the house where Jeff Thatcher lived, he saw a new girl in the garden — a lovely little blue eyed creature with yellow hair plaited into two long-tails, white summer frock and embroidered pantalettes. The fresh-crowned hero fell without firing a shot.

Tom is so enraptured by her beauty that he promptly forgets all about his previous girlfriend. Tom's first move is to act like he doesn't know Becky is watching him. Then he proceeds to show off:

he pretended he did not know she was present, and began to ‘show off’ in all sorts of absurd boyish ways, in order to win her admiration.

Readers are not told anything specific about what Tom does, other than that he was performing some kind of acrobatic gymnastics. Becky acts just as coolly as Tom and wanders back into the house without giving Tom any kind of feedback; however, just before she disappears into the house, Becky tosses a pansy over the fence in Tom's direction.

But his face lit up, right away, for she tossed a pansy over the fence a moment before she disappeared.

Tom then tries to act so cool and nonchalant about the flower that he tries to hide his picking up of the flower by picking it up with his toes:

in his efforts, he edged nearer and nearer toward the pansy; finally his bare foot rested upon it, his pliant toes closed upon it, and he hopped away with the treasure and disappeared round the corner. But only for a minute — only while he could button the flower inside his jacket, next to his heart . . .

When Tom first sees Becky in the garden, he begins to "'show off' in all sorts of absurd boyish ways, in order to win her admiration" (Chapter 3).  Becky pretends to ignore him, but before she goes inside she tosses a pansy over the fence to encourage his attention.

Soon afterwards, Tom's brother Sid breaks the sugar bowl, and their Aunt Pollly, assuming Tom did it, smacks him.  When she realizes Tom didn't do it, she neither apologizes nor reprimands Sid, but tells Tom he probably deserved the blow anyway for something else bad he most likely did when she wasn't looking.  Tom knows that "in her heart his aunt was on her knees to him" (Chapter 3), and, to comfort himself, imagines scenes of himself sick on his deathbead or being brought home dead from drowning in the river, with his aunt begging forgiveness.