It's in chapter 3 of Animal Farm that Napoleon appropriates the “nine, sturdy puppies” belonging to Jessie and Bluebell.

As with any dictator, Napoleon understands the overriding importance of catching them young—of indoctrinating one's subjects at the earliest possible age. By taking the puppies away from their mothers, he's beginning the process of their indoctrination. Napoleon intends to train them to act as attack dogs for his brutal regime. As well as forming a formidable personal bodyguard, they will also be set upon any animal that dares to step out of line.

As always with Napoleon, he isn't thinking about what's best for the animals, but only for himself. Taking the puppies from their mothers is about consolidating his dictatorship by making sure that no one gets any ideas about trying to depose him.

However, Napoleon doesn't just rely on brute force in exerting power over the animals. Through the medium of his propagandist-in-chief, he uses gaslighting—making people question the reliability of their memories—to convince the animals that it was Napoleon and not his bitter rival Snowball who was the hero of the Battle of the Cowshed.

Napoleon also stoops to outright trickery to maintain his grip on power. This can be seen in his creative rewriting of the Seven Commandments of Animalism to justify the pigs' drinking of alcohol and sleeping in beds. This is not what Old Major intended at all, but as Napoleon believes that the pigs are superior to all the animals on the farm, he's prepared to rewrite the Seven Commandments to suit himself and the other pigs.

It is in Chapter Three of Animal Farm that Napoleon takes the puppies. These "nine, sturdy puppies" belong to Jessie and Bluebell and Napoleon takes them away from their mothers as soon as they are weaned. He claims to want to educate the puppies but, later on, we see that he has trained them to act as his personal (and very violent) bodyguards.

Aside from training the puppies in this manner, Napoleon uses his cunning to seize power over the other animals. We see this most clearly at the end of Chapter Two when Napoleon distracts the other animals so that he can steal the cows' milk:

"Never mind the milk, comrades!" cried Napoleon, placing himself in front of the buckets. "That will be attended to. The harvest is more important."

By using the harvest (an event of extreme importance) as a distraction, Napoleon illustrates the depth of his natural cunning and powers of deception. It also shows that he considers the pigs to be superior to the other animals and deserving of better rations, like milk and apples. It is this which belief which drives his bid for absolute power on the farm. 

Napoleon takes the puppies of Jessie and Bluebell very early in the book - section three. This shows that Napoleon right from the beginning was thinking about how to gain power. This is an important point to bear in mind, because he was more scheming than Snowball and any other animal for that matter. Here is the text:

It happened that Jessie and Bluebell had both whelped soon after the hay harvest, giving birth between them to nine sturdy puppies. As soon as they were weaned, Napoleon took them away from their mothers, saying that he would make himself responsible for their education. He took them up into a loft which could only be reached by a ladder from the harness-room, and there kept them in such seclusion that the rest of the farm soon forgot their existence.

Now as for your second question, there were many other ways in which Napoleon sought to gain and maintain power. First, he used rhetoric. He had Squealer who would always spin things to his favor. Also because Squealer was such a great speaker, he persuaded the animals that he was always right. In fact, this was one of Boxer's mottos. 

Second, he limited the education to pigs so the other animal could not read. If knowledge is power, than he had all of it. This is why none of the animals could really create any type of revolution. The only one that could have was driven out - Snowball. 

Finally, when Napoleon had things in place, he could do whatever he wanted. At the end of the book, he turned into a human and the animals did nothing, because they could do nothing - so complete was his power. 

Napoleon takes the puppies away to give them his own brand of education in chapter 3.  When they resurface, they act as his personal police protecting him, doing away with his enemies, and allowing himto rule Animal Farm through fear tactics.

The dogs are not, however, the only way Napoleon mantains power.  Arguably, an even more effective means of attaining and maintaining his absoulte power is through propaganda.  For this, he uses Squealer to continually justify his actions to the other animals and convince them that Napoleon's changing and breaking of the commandments, as well as his other policies that are seen as contrary to Animalism, is in fact in the best interest of the animals and the farm.