Rikki-Tikki-Tavi is a bully because he does not think about the needs of others and imposes his will on them.
When people think of this story, they usually view Rikki-Tikki-tavi as a hero because he saves the family from the cobras. However, if you look carefully you will realize that he is not behaving out of love for the family but out of instinct. Also, the cobras are the original inhabitants of the area, and the family is actually invading their land. Rikki-Tikki shows no appreciation for that, and makes no attempt to find a compromise or solution where everyone can stay alive. He just kills them.
Rikki-tikki does want to be a house mongoose because all mongooses want to be house mongooses. I imagine they do. They are well taken care of that way. He does not seem to have much affection for the family though. Rikki-tikki is self-centered and arrogant. For example, he is not at all sympathetic to Chuichundra, the muskrat, when he is afraid.
"Don't kill me,'' said Chuichundra, almost weeping. "Rikki-tikki, don't kill me.''
"Do you think a snake-killer kills musk-rats?'' said Rikki-tikki scornfully.
When Chuichundra won’t tell Rikki-tikki where Nag is, he threatens to bite him. Yes, Chuichundra is taking too long and Rikki-tikki is just trying to speed him up but being nicer to him when he was afraid might have gotten results faster and more accurately.
Rikki-tikki does not ever stop to consider why Nag and Nagaina are fighting him. They are afraid of him from the moment they see him, because as far as they are concerned, a mongoose will automatically try to kill them. Also, he overhears this conversation after talking to Chuichundra.
"When there were no people in the bungalow, did we have any mongoose in the garden? So long as the bungalow is empty, we are king and queen of the garden; and remember that as soon as our eggs in the melon-bed hatch (as they may to-morrow), our children will need room and quiet …"
To Rikki-tikki, this is just another excuse to kill the entire cobra family. However, what it really means is that they are fighting because they are trying to protect their family. They are just two parents who are afraid, because their home has been invaded and their children are in danger, just like Darzee’s was. What Nag asked Rikki-tikki was a valid question. What is the difference between a snake eating a bird and a mongoose eating an egg? It is all part of nature.
There may have been some attempts at a solution where the cobras could have lived, and gotten the babies out safely. I realize this is far-fetched, but Rikki-tikki makes not attempt to even consider something. He just tries to kill them all. If the cobras had been allowed to leave with the babies, all of the death and destruction could have been avoided.
When Rikki-tikki kills all of the young cobras, it is not just to make sure that they do not grow up to be snakes. It is partly just a strategy to get Nagaina to follow him (and he chuckles when he does it). They are nothing more than pawns to him. He uses the egg in his mouth to lure her, and taunts her with it.
“What price for a snake's egg? For a young cobra? For a young king-cobra? For the last -- the very last of the brood? The ants are eating all the others down by the melon-bed.''
That is definitely rubbing it in. You can try to get her angry and get her to follow you, but there is fighting, and there is bullying, and Rikki-tikki seems to cross a line here.
In the end, Rikki-tikki does kill Nagaina, and no one will dare compete with him again in the garden. If there are other snakes, they know better to challenge him. None of the other animals will either. They are aware of his personality. He has proven the type of creature he is. He may be tough, but he is not friendly.
Although this may seem like a simple story of a valiant mongoose fighting the evil snakes, when you look deeper you will see that the snakes were actually there first, and the people and the mongoose were invading their home. In fact, when you realize that the story takes place in India, and where the British colonized, the story takes on a completely different meaning, doesn't it?