1. Why did Arnav keep asking Khushi to put on the seat belt?
The seat belt could be symbolic of Khushi's chastity (chastity belt). Khushi has been openly encouraging NK in order to make Arnav jealous and open up to his real feelings. Arnav indirectly warns her that she is taking a risk in doing so by repeatedly asking her to put on the seat belt . Although Khushi does not respond to his instruction, she indicated that she has safely tucked herself behind the seat belt, as if to show that even if she indulges in mild flirtation with NK, she will take care of guarding her honour.
2. Why did Arnav use his teeth instead of his fingers to pull out the broken piece of the bangle out of Khushi's wrist?
Arnav could see that Khushi was in a lot of pain because of the embedded piece of glass bangle. Her eyes were fixed on him as he held her wrist tightly in his hands. If he were to pull it out in front of her, she may feel the pain more. So he deliberately bent forward, making it appear as though he was going to kiss her. Intuitively she shut her eyes as she brazed herself for the contact. Having achieved his objective (of getting her to shut her eyes so that she does not have to see him pull out the glass piece), he proceeded to remove the piece with his teeth. This also gives him an opportunity to graze her wrist with his lips. So when he is pulling out the glass piece, she does not realize it, as she may have been lost in that intimate touch.
Another reason could be that the glass was embedded from the side, with most of it inside her skin/flesh. If he were to use his fingers, (which would be thicker than his teeth) he may have to grasp the visible end and yank it free which could cause her more pain. His teeth, on the other hand, could make it possible for him to slide the piece without twisting it free.
Now to the symbolic meaning
The bangle represents Khushi 's brief (non) engagement with Shyam . Although she knew Shyam was unsuitable for her she still complied with the wishes of her family. This is symbolically shown in the scene where Khushi forces the bangle down her wrist, even after realising it was small for her. In the process she gets hurt when she comes to know of Shyam's truth which led to the breaking of the engagement (bangle). One of the pieces gets embedded in her wrist. This is symbolic of Shyam's mental torture of Khushi post the break up. Arnav may employ the most unconventional methods (as seen in the use of teeth instead of hands) to get Shyam out of Khushi's life. Khushi may not realise why and when he does that. It also shows that he is willing to take the risk of hurting himself (it is more risky to remove glass piece with the teeth than with the finger because one can get one's tongue cut if one is not careful) than hurting her.
Edited to add:
Arnav buys her bangles that fit her perfectly but tries to hide the fact that he did, indicating that Arnav knows the appropriateness of their relationship and will take it forward, although he will do so stealthily and unobtrusively.
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