Shikamaru was acting like a soulless robot, devoted entirely to "The Rules". Shikamaru has never acted like that.
The fact that Tsunade, of all people, sent Kakashi off to his death is insane. Did the writers understand her character at all?! Her losses of Dan and Nawaki nearly destroyed her, and she willingly went along with a plan that would've most likely destroyed any kind of trust or loyalty Naruto and Sakura had in her?! No. Bull. kitten.
I believe Shikamaru was in character. Tsunade allowing Kakashi to go off and die did seem out of character, even though she didn't really send him, he volunteered. I know she let Jiraiya-sama go into an extremely dangerous situation vs Pain but she didn't know for sure he would die. I believe a lot of your disbelief could've been quelled had they addressed what I'll discuss about in the portion discussed below.
The Daimyo threatening to destroy Konoha because a random psychotic ninja declared war is laughable. It's the single dumbest moment I've ever seen in Naruto, and that includes fanfiction. Why would anyone believe the announcement, anyway?! Some guy nobody's ever seen before talks about starting the fourth world war, so Konoha must be planning a Coup?! What type of insane logic is that?! And don't even get me started on Kakashi willingly going to his death. He'd never do something so insanely reckless! He knows his death would hit his teammates hard, especially Naruto. There is no way in hell he'd ever go along with a plan that sacrificed a person for "The greater good."
One of the things I forgot to mention in my earlier review, is the iffy 'dropped in the middle of something' deal they gave as a starting point to the film. We were dropped into the middle of the pursuit of the villain, which may have taken months to do, which is fine, but we're never fully given a gravity to the situation. We're very hastily supposed to swallow that whomever this villain is, he's got all of Konoha frightened, and suddenly all of the world. The development of the threat is skimmed over to the end of the rising action. I say end, because we never go through the beginning and middle of the rising action where the heroes come up with a plan of attack, the trial and error, the logical journey, the thought processes of the characters. So we never find out why Kakashi feels this is the only way to end this, when he always looks for a different way out before considering death. And for them to skim over the probably thought process he goes through: plan a), plan b), etc... is what causes the loss of character believability. The Daimyo's decision to suddenly want to destroy Konoha as a result of the projection of the probably well known fugitive, also comes off as untrue because, it's one guy. How dumb does, not only the Daimyo, but the rest of the shinobi world have to be to not consider that this may be a hoax to create another war. They would at least consider that someone may have come across or simply made their own Konoha headband and thought it would be devious to pit the whole world against Konoha. They would have a meeting first and at least try to talk things out like normal politicians do, instead of suddenly believing the projection. If this were covered, the problem of character believability may be less of an issue.
Shikamaru is always the voice of reason among the rookies, and although he has been wrong, he'll do what is responsible. And actually if the main villain was as powerful as they wanted us to believe he may be I would've sided with Shikamaru. So I think he was the voice of reason as he has been in part 2. I know Naruto's team and most specifically Kakashi helped him avenge his Sensei's death, but for him to let his emotions/ sentimental duty get in the way of his better judgement when the fate of the whole village rests on Kakashi's sucess would be more out of character. That's the other thing, Kakashi specifically asked him to do it. At this point it still feels like Kakashi's actions and Tsunade's actions are more out of character than Shikamaru's. But as far as Kakashi's willingness to sacrifice himself to protect others, that's not out of charcter at all. Even if it would hit Naruto or Sakura hard. If he had to choose between protecting their feelings or protecting their lives as well as the rest of the village I don't think there's a contest there. I just don't think he would've come to the conclusion as lightly or as quickly as he does in the film.