하늘을 찌르다

은재: 흥! 잘난 척이 하늘을 찌르더니...
Meaning: Huh! And he was so full of himself!
What it actually says: His opinion of himself was so high it could pierce the skies, but now...

은재 says this about 이준 when she thinks that he cannot find his way in Venice and needs her help.

There are two expressions here: 잘난 척 and 하늘을 찌르다.

잘난 척 is from 잘난 척하다 or 잘난 체하다, which means "to think highly of oneself", "to put on an air of importance". It's a combination of 잘나다 "great, distinguished" and  척하다  (체하다) "to pretend to".
하늘을 찌르다 (" to pierce the sky") means that something is sky-high. When referring to people it can also mean that somebody is elated or energetic about something, but here the first meaning (very high) is used.
Finally, 더니 is used when something changes or is no longer true.

죽으란 법은 없다

은재 learns that she will get a scholarship that even covers her living expenses if she passes the final exam:


은재: 죽으란 법은 없다, 현성야. 어떻게 이런 일이! 감사합니다.

I came across this one in Only You textbook (from Korean through Drama series). Unfortunately the textbook is for beginners only and does not explain it, so I had to search online.

The full form is 사람이 굶어 죽으란 법은 없다 (which basically means "a person cannot die from hunger"). It means that even if you are going through hardships or struggling to make ends meet, as long as you live you can always get some food to continue living. In a more general sense it means that there is hope in a most difficult situation. Another way to say this is 산 사람 입에 거미줄 치랴 (a spider cannot spin a web in a living person's mouth)