Um... Can anyone explain me why did the part timer guy in the first chapter get fired? I r...

Probably because they're uncomfortable with him laughing at Chiaki's situation haha idk that caught me off guard as well

I think it is because he is being protective of his 'little brother.' Here is his new kid brother, who is getting bouquet orders from strangers, and the part timer who took that order, instead of being concerned or describing what the person looks like (or even noting if it was another high school student or an adult), instead is laughing about the whole situation. You notice that the 'brother' first checks to ensure that his 'kid brother' is doing all right, and asking how he feels, and once he determines that he is feeling uncomfortable about the situation, be promptly turns around and fires the part timer who is treating the whole thing as a joke. It was actually a very sweet move!