Is 'I am glad to hear that' very formal or informal phrase?

Yes, it's informal. You could consider saying 'I am very pleased to hear that', or 'Congratulations! Very well done', or 'That's fantastic to hear!' or 'That's very impressive considering the minimal time you've had to do it' or 'That's impressive indeed'.

Honestly, I would not worry about it. You have said something nice to this person, so she has no right to become offended just because it was worded informally. I'm sure she won't, though.

As it is, you were very kind indeed to compliment the professor. I'm glad to hear it!