Can I use "of" instead of "on" here?

What does she mean by "give the bird"? Why is there a quotation mark around "the bird"?

"Bird" here means middle finger! So "to give the bird" means the same thing as "to give/flip the [middle] finger." Ex: He gave the bird to the driver that cut him off. Ex: I got in trouble with my teacher for flipping the bird to a classmate.

Please tell me the structure of the sentence. Are subject and object reversed?

This sentence is considered an inversion sentence. An inversion sentence is when the verb is put before the subject. Since this sentence starts out with a prepositional phrase, the verb is "comes" and the subject has to be the puppy. "A whole new world of sensory stimulation comes with this," would be a more common way to say this sentence. However, the narrators decided to put the adverbial expression (And with this) in the beginning of the sentence. Doing this makes the sentence sound more formal.

Can I say "she seemed to be starting" instead of "I reckon she was starting"?

"I reckon" is a more formal way to say "I think" or "I believe". You could say "she seemed to be starting" instead, however, it would take away the personal idea of the speaker. Therefore it would be better to say "I think she was starting to annoy you" instead.

what does "have/has been around" mean?

In this case "Has been around" means (1) "to have existed". It can also mean (2) "to have a lot of experience in something" Ex: The Bible has been around for thousands of years. Ex: Jim has been around for a long time. In fact, he joined our club ten years ago. Ex: Look, I've been around and I've bought a lot of jewellery, I know a real diamond when I see one. Ex: Bob knows how things work. He's been around HayanMind for a long time.

What does "like" mean here?

Yes, you can ignore "like" here when you translate this sentence. A lot of Americans have a habit of saying "like" a lot as a filler word, and often use it when they don't know what to say. When "like" is used in this way, it does not add any meaning to the sentence.