Why did Kakashi not kill Hidan? A few episodes after the battle with Hidan and Asuma... :'(. Shikamaru, Ino, Choji and Kakashi set out to find Hidan and Kakuzu. When they find them Shikamaru uses his shadow jutsu to "paralyze" the enemies. So far they can only see just Shikamaru. Why doesn't Kakashi use his new sharingan to completely obliterate hidan while he was frozen?

Please no spoilers as I am only in mid season 2.
THAT IS STRATEGY. When you don't even know a single thing about the enemy, would you go face to face directly? Well, in Shikamaru's case, there must be a person to be the bait, and once they make sure that the enemies are completely immobilized, the plan begins. Unfortunately, Kakuzu freed himself, so the plan failed.
Why did Ino did not use the Mind Body Switch Technique on Kakuzu?
There are several reasons why Ino decided not to use it.
1. If the opponent's will is strong enough, they can force the user to leave their body. Apparently the stronger one has the stronger will.
2. If the user's chakra level is low, the technique will be harder to maintain.
3. If any physical injury is inflicted on the victim's body while the mind link is active, it would cause the user's body to also receive those same injuries. This is a good chance for Hidan to kill her by piercing through one of Kakuzu's hearts, and Kakuzu will still be fine since he has multiple hearts. That is not a good idea to eliminate the target.
Why didn't Choji, Ino and Shikamaru apply Formation Ino–Shika–Chō in the battle to distract Kakuzu while Kakashi was aiming at Hidan with his Mangekyō Sharingan?
1. Kakashi initially having trouble to aim at his target with Kamui properly, he struggled to hit his target. Recall the episode when Kakashi is using Kamui on Deidara. Hidan is shown to be very skilled in melee combat, meaning he could run and jump to dodge Kakashi's aim.
2. It requires some amount of preparation time and has a huge drain on his chakra each time it is performed. If he missed, a huge portion of his chakra is wasted.
3. Kakashi hasn't completely mastered Kamui yet, so at the time transporting an arm was his limit. Don't forget that Hidan is an immortal bastard, decapitation only makes him immobilize. Even though Kakashi was capable to aim at Hidan's head accurately, Kakuzu could reveal his 4 masks with different nature affinity to disturb Kakashi.