In the clip I'm analysing, only 3 shots are used:
The first one is a long shot of Toothless (his full body is in shot) with selective focus (Hiccup is slightly out of focus). It is clear in this shot that the audience should be focusing on Toothless and his reaction to Hiccup.
The second is a medium shot of Hiccup (shows him from waist up) to focus on his reactions and thinking. He's playing a bit of a dangerous game here by standing face to face with a dragon and the audience needs to see how he reacts and what he's thinking. This shot allows the audience to focus on his facial expressions and actions with the fish when necessary.
The third is another long shot but of Hiccup (full body in shot). It's a longer shot than normal because of the space left around Hiccup's body. It's not enough to be an extreme shot but it establishes something important to the audience; how in the open Hiccup is. This shot is used when Hiccup is forced to drop his knife away from his body. Being out in the open with a dragon is dangerous enough but this shot emphasises how Hiccup, on the act of throwing away the knife, is truly alone with nowhere to run and now completely defenseless.
There could be a fourth shot, which is the slight close up on Hiccup's face (from shoulders up). This is only used once in this clip to really focus on Hiccup's response to Toothless' disapproval of him having the knife. His facial expressions are key to his thoughts. In this shot we can see that he decides, knowing its dangerous, to throw it away.
The shots are kept minimal and simple. There is no dialogue in this scene but there is a 'conversation' between the two characters through body language alone, which is why it is important to have Toothless' whole body in shot. The cuts in this scene are timed so that after a character does his action, we see the other character's responding action and then back again to see the first character's response to that, creating this 'conversation'.
The main focus here is on the characters building a relationship.
The rule of thirds is used in this clip to keep the shots interesting but more importantly to clearly show that these characters are interacting by spacing them appropriately when both in shot at the same time.
Toothless is never shown alone in the shot, Hiccup is always out of focus in the foreground. However, Hiccup is shown in shot alone but slightly off centre towards to left. He is always looking at something downwards towards the right. I think this is intentional because if Toothless was kept in the shot there wouldn't be enough focus on Hiccup and his responses and thoughts.
In this last screen capture he is a bit farther on the left. He is still looking at something downward towards the right. With all the space around him. he is giving the impression that the character on the other side of the conversation is the dominant one and making him feel small. He does look quite small in that shot!