There are paid internships and unpaid internships. You can't guess from the word itself. You have to ask.
"A monster of a day" can mean that it was a very big day - full of good things or bad things. Based on the context, it seems to be a very bad day.
There was a monster of a sale at the shoe store. -- A lot of things were on sale and the sale was big. Good thing.
There was a monster of a car crash. -- There was a crash involving many cars. Bad, bad thing.
Student or Learner