Why does Todoroki look different between the 1st and 2nd season?
In the first picture, it was Todoroki wearing his hero costume. In episode 6 to 8, they had mock battles wearing their costumes. It is also explained that the students can give their own design for their costume.

The second one, it's in the Sports Festival where they cannot wear hero costumes because they will have an advantage to their enemies.



Regarding freezing and shivering, if you're referring in the 1st picture, no, he won't freeze because that is just his hero costume, not his quirk at work.

But if you're talking about his quirk, then the answer is still no, because he uses the other half of his quirk which is 


   half hot


to keep himself from the effects of his half cold and vice versa.