Tobirama trusted Kagami because he knew he was one of the Uchiha who had conquered the clan's curse of hatred. Hiruzen, on the other hand, was clearly the most talented on the squad, as well as the most at comfortable with himself, since he was the only one to offer himself as a decoy, and he did it quietly. Thus, it can be said that they were equal partners in crime.
However, this does not mean that they were friends. They probably met once or twice during their missions together but they didn't know each other very well. Kagami was always calm and collected when they were doing something dangerous or important; while Hiruzen would often get nervous and distracted if anything unexpected happened. This is probably why Tobirama chose him for this mission instead of himself.
As for why Kagami volunteered himself... maybe he just wanted to prove to Tobirama that he was able to take care of himself? Or maybe he just liked doing things behind the scenes and felt like helping out was his duty as the partner of the Uchiha who had defeated the clan's curse? We'll never know unless he tells us himself!
In conclusion, yes, Kagami hated Tobirama. He blamed him for the death of his wife and son and wanted to kill him with his own hands.
Shisui Uchiha's predecessor was Kagami Uchiha. He was Danzou and Hiruzen's teammate. He was one of countless in whom Tobirama had complete faith. Tobirama had a six-man group reporting directly to him. The other five men were: Iwagakure's Tsunade, who was once called "Danzō", but now goes by the name "Kagami"; Hanzō Isamu, a ninja from Iwagakure; Hyūga Neji, a former assassin for hire; Yamanaka Gennosuke, a master thief; and Kurotsuchi Shibata, a former member of Danzo's gang.
Later on, after Shishio's death, Kagami became the new head of the Tobi clan. However, he was soon killed by Shino Aburame who wanted to kill the only remaining person who knew her secret identity as Kaede Fuyuhiko. After his death, Kaede decided to use his knowledge to travel around the world in order to find more allies like herself. She eventually found two people who showed interest in joining her cause and gave them new names - Hashirama and Sayo. Together, they created a new clan called Sandaime Hokage which means "Deputy Chief Executive Officer" or "Vice President".
Now, about Shino Aburame.
This is one of the key reasons he works so hard. Oikawa also has a vendetta against Kageyama, whom he mockingly refers to as "Tobio-chan." This is owing to his sentiments of inferiority toward Kageyama, who has been acclaimed as a genius setter since their time at Kitagawa Daiichi. Oikawa believes that because Kageyama is a setter, he should be doing something easy like setting up traps, not thinking up strategy games. He feels that if he were a true genius, he would be able to come up with ideas for new games on his own, but because he can't, he has no choice but to work under Kageyama.
There is another reason why Oikawa hates Kageyama: they are both samurai. Oikawa takes great pride in being a samurai, and he sees Kageyama as an obstacle to overcome in this quest for greatness. He believes that by defeating Kageyama, he will prove that he is a better samurai than him.
However, what most people don't know is that during the war between the Tokugawas and the Hojo, Oikawa sided with the Tokugawas. When Kageyama betrayed them and joined the Hojos, Oikawa followed suit. This shows that even though Oikawa thinks he's better than Kageyama, in fact, he is only as good as Kageyama.
Kimimaro remained faithful to Orochimaru even after being resurrected by Kabuto and wanted to carry out Kabuto's directives. Gaara remarked on Kimimaro's hastiness and his dislike for inferior opponents, or "trash" ninjas, as he described it. However, when given the chance to kill Sasuke Uchiha, who had just defeated him, Kimimaro refused due to his loyalty to Orochimaru.
When told by Kabuto that he has been chosen to assassinate Hanzou, Kimimaro refuses again, this time because he believes that killing someone while they are asleep is wrong. However, when Orochimaru reappears and tells them that killing people while they are sleeping is not only acceptable but also an important part of his plans, Kimimaro agrees to do as commanded.
Orochimaru uses his powers to make Kimimaro believe that his wife and son are dead. When asked if he still wants to follow Orochimaru, Kimimaro replies that he does not have any choice in the matter. Not only that but he also gets brainwashed into thinking that Gaara is his enemy too.
In the end, Kimimaro was freed from Orochimaru's influence by Nagato who used his Rinnegan to erase everyone's memories of Kimimaro following Orochimaru's orders.