Our professor is allowing us to bring a formula sheet into the exam with us. Can I add other notes to help me to remember the steps to solving equations?

No. You are only allowed to bring in approved test materials. depending on the professor and the course, these might include a page of notes, a list of formulae, your text book, or something else. No matter what is allowed, you are not permitted to supplement those things with additional notes or information.

My friend has a copy of the solution manual for my chemistry class. It makes it a lot easier to get the homework done!

Unless your instructor has approved the use of a solutions manual, you are cheating. You are also limiting how well you will learn the material.

Hey, Jazmine just asked me for help because her dog's been sick. I like her a lot, so I'll just let her copy my answers.

  It's cheating if you let anyone copy your answers because you are helping someone cheat!