... なくてはいけない means have to do ...
when it is used after 行く (go) , it becomes 行かなくてはいけない .
In speech, the ては in the above form often becomes ちゃ , so it becomes 行かなくちゃいけない.
なる can be translated as becomes。
行かなくちゃいけない + なる --> 行かなくちゃいけなくなる
It literally means:
I "becomes" (to be in a state that I) "have to do ...",
which in this sentence you can translated it into I was asked to an interview ...
The past tense of the above form is 行かなくちゃいけなくなった。
The final の is only a particle (for emphasis, maybe).
So: なくてはいけない(なくちゃいけない)+なる+た+の ==> なくちゃいけなくなったの
For the second, this should be syntactically wrong. But you can hear this form in real speech. It means "if you do not leave a good first impression". I think you can treat it as 第一印象をよくしないとね. と here is an particle shows an assumption, and the latter part (the "then" part) is omitted. ね is a particle with no real meaning.